I spy the confluence of mathematics, psychology, and physics

I find the relationship between mathematics and vision fascinating.   Even within mathematics itself, seeing how the geometric expression of ideas can clarify or further develop countless mathematical thoughts is always worth noting – like the graphs of functions, or the projections of figures.  I’ve written before about the relationship between the brain’s visual processes and mathematics.  And, along these lines, I had reason to look a little further into Hermann von Helmholtz’s contributions to both vision and mathematics.

Nineteenth century France and Germany broke from past ideologies, and new economic and political structures emerged.  There were significant developments in science and mathematics and significant growth in specializations.  I’ve highlighted the work of Bernhard Riemann often, paying particular attention to his famous 1854 paper On the hypotheses which lie at the bases of geometry, to some extent because Riemann acknowledged the influence of philosopher Johann Frederich Herbart who pioneered early studies of perception and learning.   I wrote a piece for Plus Magazine that suggested parallels between Riemann’s insights into the nature of space, quantity and measure in mathematics, and modern studies in cognitive neuroscience that address how number, space, and time are more the result of brain circuitry than features of the world around us.

It became particularly clear to me today that another nineteenth century heavyweight, whose multidisciplinary research spans physics, psychology, and mathematics, was similarly influenced by Herbart.  In an essay with the title The Eye as Mathematician, Timothy Lenoir discusses Hermann von Helmholtz’s mid-nineteenth century theory of vision, which suggests an intimate link between vision and mathematics. And Lenoir explains that Helmholtz’s theory of vision was “deeply inspired by Herbart’s conception of the symbolic character of space.”

Lenoir sketches out how the brain uses the data it receives to construct an efficient map of the external world. The data may include sensory impressions of color, or contour along with, perhaps, the registration of a light source on a peripheral spot on the retina.  The location of the light source is then defined by the successive feelings associated with eye movements that bring the focal part of the eye in line with the light. A correspondence between the arc defined by each positional change in the eyes, and the stimulation of that spot on the retina, is stored in memory and repeated whenever that spot on the retina is stimulated.  Helmholtz called these memories local signs.  They are learned associations among various kinds of sensory data that also include head and body positions. From sequences of sensory inputs, the mind creates pictures, or symbolic representations, that provide a practical way for us to handle the world of objects we find around us. Helmholtz is clear, however, that these pictures or symbols are not copies of the things they represent. While causally related to the world around us, the quality of any sensation belongs to the nervous system. For Helmholtz, the things we see are a symbolic shorthand for aggregates of sensory data like size, shape, color, contrast, that have become associated through trial, error and repetition. The more frequently associations occur, the more rapidly linkages are carried out. Symbols then become associated with complexes of sensory data. And, like a mathematician, the brain learns to operate with the symbols instead of with the production of the complex of sensory data directly. This, Helmholtz argued, is how the constructive nature of perception becomes hidden and nature seems to us to be immediately apparent.

There are other psychological acts of judgment in Helmholtz’s visual theories. The brain has to decided whether a collection of points, for example, generated by stimulation of the retina, does or does not represent a stable object in our presence. To be an object, the points registered on the retina would need to be steady, to not move or change over time. The brain tests their stability by evaluating successive gazes or passes over the object. According to Helmholtz, the collection of points is judged to be a stable object if the squares of the errors, after many passes, are at an acceptable minimum. This is meant in the same sense as the principle of least squares in mathematics. Lenoir calls these measuring mechanisms sensory knowledge, “part of the system of local signs we carry around with us at all times…”

Lenoir’s piece also made it clear that, in the mid 1800’s, there was significant overlap in the methods and the instruments developed by physiologists and astronomers.  Gauss introduced the use of least squares in astronomy.  Helmholtz invented the ophthalmometer, an instrument that measures how the eye accommodates changing optical circumstances, which makes the prescription of eyeglasses possible.    He described the ophthalmometer as a telescope modified for observations at short distances.

In an article for the Stanford Encyclopedia of Philosophy, Lydia Patton also addresses Helmhotz’s work in mathematics.

Even when he was writing about physiology, Helmholtz’s vocation as a mathematical physicist was apparent. Helmholtz used mathematical reasoning to support his arguments for the sign theory, rather than exclusively philosophical or empirical evidence. Throughout his career, Helmholtz’s work is marked by two preoccupations: concrete examples and mathematical reasoning. Helmholtz’s early work in physiology of perception gave him concrete examples of how humans perceive spatial relations between objects. These examples would prove useful to illustrate the relationship between metric geometry and the spatial relations between objects of perception. Later, Helmholtz used his experience with the concrete science of human perception to pose a problem for the Riemannian approach to geometry.

As I read, I felt like I was enjoying just a little sip of the rich confluence of physics, psychology and mathematics. We keep trying to unravel the tight weave that binds the nature of the world, the nature of our perception and experience, and how we pull it all together.

Locating Meaning

Recently I became particularly sensitive to discussions that address ‘meaning’ as an emergent property of both biological and formal systems (of which mathematics is one). And this is because it is the meaning of symbols in mathematics that is the source of its power. But it is not at all clear that the meaning of mathematical symbols is purely the meaning that we attribute to them. Meaning is not just assigned to mathematical symbols. It seems that meaning also emerges from them.

In an article that appeared in a 2006 issue of the Bulletin of Mathematical Biology,  immunologist Irun Cohen argues that meaning is not an intrinsic property of an entity but rather emerges from dynamic systems. Cohen’s article was used in last month’s post to explore the idea that information in biological systems feeds back on itself in such a way that modified copies of old information, enrich the system with new information, assuming that these modified copies are not destructive to the old information or to the system in general. For Cohen:

Meaning, in contrast to information, is not an intrinsic property of an entity (a word or a molecule, for example); the meaning of an entity emerges from the interactions of the test entity (the word or molecule) with other entities (for example, words move people, and molecules interact with their receptors, ligands, enzymes, etc.). Interactions mark all manifestations of biological structure—molecular through social. Meaning can thus be viewed as the impact of information—what a word means is how people use it; what a molecule means is how the organism uses it; what an organism means is what it does to others and how others respond to it; and so on over the scales life—meaning is generated by interaction.

As last month’s post explained, the ideas expressed in this article are linked to the work of biophysicist and philosopher Henri Atlan. Much of Atlan’s work is directed at understanding the mechanisms of self-organization in systems that are not goal oriented from the outside. Instead, these systems organize themselves in such a way that the meaning of information emerges from the dynamics of the system.

These ideas brought Douglas Hofstadter’s famous text, Gödel, Escher, Bach, to mind again. Hofstadter spends a significant amount of time asking questions about the location of meaning in order to establish “the universality of at least some messages,” or some information. Meaning, he argues, is an inherent property of a message, if the context that gives it meaning is so natural that it is part of the universal scheme of things.  Or, it’s so natural, that it’s everywhere.

It turns out that locating meaning is not a simple matter. In the case of a vinyl recording, for example, we can ask whether the meaning is in the grooves of the record, or in the sound waves produced by the needle on the grooves, in the brain of the listener, or in what the listener has learned about music? In mathematics, is the meaning of symbols coming from chains of human experiences – like collecting and sorting – that are linked by metaphors? Or is it coming from the way relations among abstract objects mirror cognitive processes. Or, is it coming from our immersion in the universal properties that they express? Mathematics could look like a game, where the rules are made to establish relations among symbols. To an untrained eye, it all looks fairly arbitrary. But it’s not. And locating its meaning is, perhaps, the way we understand how it is not a game. This is important to our understanding ourselves.

In the preface to the 20th anniversary edition of Gödel, Escher, Bach Hofstadter argues that patterns, bring about our self-reflective consciousness – the very thing that is at the heart of mathematical systems:

…the key is not the stuff out of which brains are made, but the patterns that can come to exist inside the stuff of a brain. This is a liberating shift, because it allows one to move to a different level of considering what brains are: as media that support complex patterns that mirror, albeit far from perfectly, the world, of which, needless to say, those brains are themselves denizens – and it is in the inevitable self-mirroring that arises, however impartial or imperfect it may be, that the strange loops of consciousness start to swirl.

It is a particular kind of pattern that Hofstadter has in mind, something he calls a strange loop – patterns that refer back to themselves. While Atlan and Hofstadter are not actually saying the same thing, there is certainly some overlap between Atlan’s focus on self-organizing systems and Hofstadter’s use of self-referencing systems. And so there is no surprise, perhaps, when the Fluid Analogies Research Group (which Hofstadter heads) describes ‘thinking’ as

…a kind of churning, swarming activity in which thousands (if not millions) of microscopic and myopic entities carry out tiny “subcognitive” acts all at the same time, not knowing of each other’s existence, and often contradicting each other and working at cross-purposes. Out of such a random hubbub comes a kind of collective behavior in which connections are made at many levels of sophistication, and larger and larger perceptual structures are gradually built up under the guidance of “pressures” that have been evoked by the situation.

In Gödel, Escher, Bach, Hofstadter argues that Euclid actually obscured the paths that geometric ideas could open by allowing the real world meaning of words like point, line, and circle to persist in his formal system of deductive reasoning. As a result he explains, “some of the images conjured up by those words crept into the proofs which he created.”  It’s a subtle effect, but that’s what’s interesting about it. And hence all of the proofs that attempted to confirm Euclid’s facts about parallel lines were inevitably contaminated by the interplay of everyday intuition and the formal properties of an abstract system. The meaning of objects and propositions in this system did not actually reside in experience. According to Hofstadter:

By treating words such as “POINT” and “LINE” as if they had only the meaning instilled in them by the propositions in which they occur, we take a step toward the complete formalization of geometry.

This opens many doors to understanding, not the least of which are the non-Euclidean geometries.

Gödel, Escher, Bach uses mathematics to address the emergence of the self-reflective ‘I’ in our experience, and Gödel’s theorems are at the heart of the matter. The fact that Gödel’s numbering made it possible for mathematics to make a statement about itself was Hofstadter’s inspiration for Gödel, Escher, Bach and the research efforts that followed. It always looked to me like mathematics was alive. I just find more and more reasons to think I was right.

Spaces

I read about the sad passing of Maryam Mirzakhani in July, and the extraordinary trajectory of her career in mathematics. But I did not know much about what she was actually doing. A recent post in Quanta Magazine, with the title: Why Mathematicians Like to Classify Things, caught my attention because the title suggested that the post was about one of the most important ways that mathematics succeeds – namely by finding sameness among diversity. I found that the work discussed in this post addresses the mathematical world to which Mirzakhani has made significant contributions. Looking further into the content of the post and Mirzakhani’s experience invigorated both my emotional and my intellectual responses to mathematics.

A Quanta article by Erica Klarreich was written in 2014, when Mirzakhani won the Fields Medal. There Klarreich tells us that when Mirzakhani began her graduate career at Harvard, she became fascinated with hyperbolic surfaces and, it seems, that this fascination lit the road she would journey. These are surfaces with a hyperbolic geometry rather than a Euclidean one.  They can only be explored in the abstract. They cannot be constructed in ordinary space.

I find it worth noting that the ancestry of these objects can be traced back to the 19th century when, while investigating the necessity of Euclid’s postulate about parallel lines, mathematicians brought forth a new world, a new geometry, known today as hyperbolic geometry.  This new geometry is sometimes identified with the names of mathematicians János Bolyai and Nikolai Ivanovich Lobachevsky.  Bolyai and Lobachevsky independently confirmed its existence when they allowed Euclid’s postulate about parallel lines to be replaced by another.  In hyperbolic geometry, given a line and a point not on it, there are many lines going through the given point that are parallel to the given line.  In Euclidean geometry there is only one.  With this change, Bolyai and Lobachevsky developed a consistent and meaningful non-Euclidean geometry axiomatically.  Extensive work on the ideas is also attributed to Carl Friedrich Gauss. One of the consequences of the change is that the sum of the angles of a hyperbolic triangle is strictly less than 180 degrees.  The depth of this newly discovered world was ultimately investigated analytically.  And Riemann’s famous lecture in 1854 brought definitive clarity to the notion of geometry itself.

With her doctoral thesis in 2004, Mirzakhani was able to answer some fundamental questions about hyperbolic surfaces and, at the same time, build a connection to another major research effort concerning what is called moduli space. The value of moduli space is the other thing that captured my attention in these articles.

In his more extended piece for Quanta, Kevin Hartnett provides a very accessible description of moduli space that is reproduced here:

In mathematics, it’s often beneficial to study classes of objects rather than specific objects — to make statements about all squares rather than individual squares, or to corral an infinitude of curves into one single object that represents them all.

“This is one of the key ideas of the last 50 years, that it is very convenient to not study objects individually, but to try to see them as a member of some continuous family of objects,” said Anton Zorich, a mathematician at the Institute of Mathematics of Jussieu in Paris and a leading figure in dynamics.

Moduli space is a tidy way of doing just this, of tallying all objects of a given kind, so that all objects can be studied in relation to one another.

Imagine, for instance, that you wanted to study the family of lines on a plane that pass through a single point. That’s a lot of lines to keep track of, but you might realize that each line pierces a circle drawn around that point in two opposite places. The points on the circle serve as a kind of catalog of all possible lines passing through the original point. Instead of trying to work with more lines than you can hold in your hands, you can instead study points on a ring that fits around your finger.

“It’s often not so complicated to see this family as a geometric object, which has its own existence and own geometry. It’s not so abstract,” Zorich said.

This way of collapsing one world into another is particularly interesting.  And one of the results in Mirzakhani’s doctoral thesis concerned a formula for the volume of the moduli space created by the set of all possible hyperbolic structures on a given surface.  Mirzakhani’s research has roots in all of these – hyperbolic geometry, Riemann’s manifold, and moduli space.

Her work, and the work of her colleagues, is often characterized as an analysis of the paths of imagined billiard balls inside a polygon. This is not for the sake of understanding the game of pool better, it’s just one of the ways to see the task at hand.  Their strategies are interesting and, I might say, provocative . With this in mind, Hartnett provides a simple statement of process:

Start with billiards in a polygon, reflect that polygon to create a translation surface, and encode that translation surface as a point in the moduli space of all translation surfaces. The miracle of the whole operation is that the point in moduli space remembers something of the original polygon — so by studying how that point fits among all the other points in moduli space, mathematicians can deduce properties of billiard paths in the original polygon.   (emphasis added)

The ‘translation surface’ is just a series of reflections of the original polygon over its edges.

These are beautiful conceptual leaps and they have answered many questions that inevitably concern both mathematics and physics.  In 2014, Klarreich’s article captured some of Mirzakhani’s thoughtfulness:

In a way, she said, mathematics research feels like writing a novel. “There are different characters, and you are getting to know them better,” she said. “Things evolve, and then you look back at a character, and it’s completely different from your first impression.”

The Iranian mathematician follows her characters wherever they take her, along story lines that often take years to unfold.

In the article she was described as someone with a daring imagination.  Reading about how she experienced mathematics made the nature of these efforts even more striking.   There is a mysterious reality in these abstract worlds that grow out of measuring the earth.  The two and three dimensional worlds of our experience become represented by ideals which then, almost like an Alice-in-Wonderland rabbit hole, lead the way to unimaginable depths.   We find purely abstract spaces that have volume.  We get there by looking further and looking longer.  I feel a happy and eager inquisitiveness when I ask myself the question: “What are we looking at?”  And I would like to find a new way to begin an answer. It seems to me that Mirzakhani loved looking.  A last little bit from Klarreich:

Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. “Months or years later, you see very different aspects” of a problem, she said. There are problems she has been thinking about for more than a decade. “And still there’s not much I can do about them,” she said.

 

 

Self-organizing, art, and mathematical mutants

Deciphering the principles of self-organizing systems is often at the heart of new ideas in biology, including neurobiology. A complex, self-organizing system contains a large

number of elements that have predictable, local interactions with each other, but these local interactions create global properties that cannot be predicted from even the most well-understood local events. This is why mechanical models of these systems fail. Flocks and swarms illustrate this kind of configuration. I’ve written before about how bees and ants optimize foraging routes, and how their optimization solutions have contributed to problem solving strategies in computer science. I’ve also posted about physicists’ observation that birds in a flock followed the lead of precisely their seven closest neighbors, regardless of the density of the flock. The rippling effect that this had on the flock resembled the physics of magnetism. In other words the birds aligned with their neighbors the way the electron spin of particles aligns as metals become magnetized.

The collective behavior of self-organized biological systems, like ant and bee colonies, has come to be called swarm intelligence. The complexity of the group’s behavior is one that could never be managed by an individual member. Nor is it directed by any individual members. Schools of fish, flocks of birds, and herds of animals, all display swarm intelligence, where the actions of individual members of are determined by an inherent set of rules, and these actions transform the behavior of the group. To many observers, this kind of collective behavior indicates that members are assembling, distributing, and processing information. This focus on the role played by information has inspired a good deal of multidisciplinary study.

While doing a little research on swarm intelligence, I came upon an article published in 2006 in the Bulletin of Mathematical Biology, and written by an immunologist whose research on the immune system involves the properties of self-organization. The author is Irun R. Cohen and the title of the article is Informational Landscapes in Art, Science, and Evolution. Cohen uses a multimedia work of art called Listening Post as “a prototypic example for considering the creative role of informational landscapes in the processes that beget evolution and science.”  The work Cohen evaluates is one that relies on self-organizing principles.

There is a trail of thoughts that lead to Cohen’s argument. These include the provocative idea that new meaning can be created from what is often thought of as interference or ‘noise’ in a signal. In other words, new meaning can be created from unstructured, random, or meaningless signals or representations. One of the keys to this possibility comes from a principle of self-organization whose formulation Cohen attributes to a colleague – biophysicist and philosopher Henri Atlan. In the context of a discussion of evolution and natural selection, Cohen summarizes Atlan’s view.

Atlan’s argument goes like this: Existing information first generates surplus copies of itself, which happens regularly in reproducing biological systems. The surplus copies can then safely undergo mutations, and so create modified (new), added information without destroying the untouched copies of the old information. The system thus becomes enriched; it now contains the new information along with the old information. Indeed, it appears that the complexity of vertebrate evolution was preceded and made possible by a seminal duplication of the ancestral genome…

..Information, in other words, feeds back on itself in a positive way; a great amount of information, through its variation, leads to even more information. And as information varies it increases, and so does complexity.

This, I think, is a beautiful idea. Cohen then shows us how the artwork Listening Post shares the features of an organism, and he explains its two sides – a visual-auditory display designed by an artist whose content is driven by an algorithm developed by a mathematician. But this is how it behaves:

The algorithm randomly samples, in real time, the many thousands of chats, bulletin boards, and bits of message that flow dynamically through the cyberspace of the Internet. This simultaneous me ́lange of signals, in the aggregate, is meaningless noise. The algorithm, by seeking key words and patterns of activity, artfully exploits this raw information to construct patterns of light, sound, and words that please human minds. The substrate of information flowing over the Internet is in constant flux so the patterns presented by Listening Post are unpredictable at the fine microscopic scale; but at the macroscopic scale of sensible experience, Listening Post is manifestly pleasing…Listening Post transforms the Internet’s massive informational landscape into a comprehensible miniature. Two attributes of Listening Post illustrate our theme: the work feeds on information designed for other purposes and it survives by engaging our minds.

Cohen develops precise definitions of information, signal, noise, and meaning. These are necessary to the clarity of his broad parallels, like this one addressing the informational structure of the cell and the internet:

In place of electromagnetic codes generated by computer networks, the information flowing within and through the cell—life’s subunit—is encoded in molecules. But the informational structure of both networks, cell and Internet, is similar: Each molecule in a cell, like a chat box signal, emerges from a specific origin, bears an address, and carries a message. Our problem is that the cell’s molecules are not addressed to our minds, so we don’t understand them. The mind exists on a different scale than does the cell; the mind and the cell live in different informational landscapes. We are unable to directly see molecular information; we have to trans- late the cell’s molecules and processes into abstract representations: words, numbers, and pictures…The informational landscape of the cell-organism-species-society is like the informational landscape of the Internet; viewed in the aggregate it is incomprehensible noise.

In his quest to understand, Cohen argues that computers can help. But specifications and the ordering of data are not enough. Understanding is not simply representation. Understanding is primarily an interaction with information. In general we understand complex information by transforming it into a meaningful signal, whether it be meaningful images, symbols, sounds, actions, etc… And this is what Listening Post does.

Part of the point of this article is to explain a research strategy Cohen and colleagues have developed, called Reactive Animation, that they hope will bring them closer to this ideal. He sees it as a synthesis of biology and information science, “between mind and computer.” He and his colleagues have developed a way to record and catalog complex scientific data and “have the data themselves construct representations that stimulate human minds productively.” The effort sounds rich and promising. But what I find most intriguing about these information driven, self-organizing ideas is the view that information breeds information, generating diversity and greater complexity. And productive offspring would be information that breeds information captivating to the human mind. I couldn’t help but consider that mathematics itself is just such a system.

Ideas have crept into mathematics that can look like aberrant variations of other things. The imaginary unit, for example, defined as the square root of negative one, is one such mutation. There is no number which when multiplied by itself will produce -1. It looks like a mistake. This odd use of number symbols showed up when 16th century Italian mathematicians employed a quirky algebraic trick to find the solutions to some cubic equations. The scheme involved the use of numbers that included the square roots of negative numbers. While it was agreed that these roots had no meaning, they nonetheless made it possible to extract the real number solutions that were sought. But these anomalies held the attention of mathematicians through the 18th century. Gauss was one among many who defended their value and explored their meaning. Eventually they produced an entirely new set of numbers – the complex numbers – made from a real number and some multiple of the imaginary unit. These numbers were not only acceptable, they produce beautiful results in mathematics and are extraordinarily useful in physics and engineering. The complex number found a home on the complex plane and produced the branch of mathematics called complex analysis. The system has certainly been enriched.

This may be one of the easiest parallels to draw, but there are others like Weierstrass’ monster function that is now understood by chaos theory and its relationship to fractals. Or, infinitesimals, once a hindrance to the acceptance of the calculus and now the foundation of non-standard analysis. Or the clarifications in geometry provided by the oxymoron: the point at infinity.   

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

 zeta(n)=sum_(k=1)^infty1/(k^n).

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.