Mathematical hybrids and the like

My attention was just recently brought to the work of philosopher and poet Emily Grosholz.  It’s rare to find an individual so steeped in the ways of poetry and mathematics, and the desire to explore how and what they express about us.  What I would like to consider here, in this particular post, is really a detail of the extensive thought and research that Grosholz brings to the discussion of how mathematics grows.  But I think it’s a powerful idea that can have a good deal to say about how we work, and how we, as a species, produce the bountiful and variegated products of human culture.

Grosholz is the author of many books that include works on the philosophy of mathematics as well as works of poetry. Her latest is Starry Reckoning: Reference and Analysis in Mathematics and Cosmology.  What follows is based on a piece that she contributed to a book she edited with Herbert Breger. The book is called The Growth of Mathematical Knowledge, and her piece is given the title The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge.  Here Grosholz argues that unlike what has been considered before, different branches of mathematics do not reduce to other branches.  Philosophers of mathematics have discussed the possibility that geometry can be reduced to arithmetic, arithmetic to predicate logic, and arithmetic and geometry to set theory. This is understood in much the same way that one might claim that biology can be reduced to chemistry and chemistry to physics.  The vocabulary of the reduced theory is redefined in terms of the reducing theory.  In the sciences, the reducing theory has been thought to play an explanatory role, suggesting an inherent unity among the various scientific disciplines.  But in mathematics the so-called reducing theory is not used so much as an explanation of the reduced ideas, but more as a foundation for them.  And mathematicians have long had difficulty with foundational questions.   Grosholz, on the other hand, proposes that mathematics is a collection of rationally related but autonomous domains and then highlights the potent role of what she calls mathematical hybrids.

She explains that in Greek mathematics the autonomy of domains is clear.  Geometry is about points, lines, planes, and figures, and geometric problems involve relations between parts of the whole of spatial figures.  Arithmetic is about numbers, and problems in arithmetic involve monotonic, discreet succession.  The vocabulary of logic is one of terms, propositions, and arguments, and problems in logic involve ideas of inclusion, exclusion, consistency, and inconsistency.   While these separate domains may seem to resist assimilation, 17th century mathematics introduced some unifications.  Among these unifications is Descartes’ application of algebraic techniques to geometric constructions, and Leibniz’s application of combinatorics to an analysis of curves.  Grosholz spends some time on each of these.  She points out that Leibniz was fascinated with formal languages and number theory, and that he believed that the art of combinations was central to the art of discovery.  She argues that Leibniz’s investigation of algebraic forms in the calculus is grounded in “an imperfect but suggestive analogy between numbers and figures.”  The infinite summing of infinitesimal differences, that becomes the integral, emerges from his ability to bridge geometric ideas about a curve (like tangent, arc length, area), with algebraic equations, and through the notion of an infinite-sided polygon approximating the curve, patterns of integers were also connected.  Here the mathematical hybrid emerges: an abstract structure that rationally relates different domains in the service of problem solving.  On a deeper level, objects in each domain must actually exhibit features of both domains, despite the instability created by their differences. But, Grosholz argues, this instability does not mean that hybrids are defective.  They are held together by the clarity of the domains from which they emerge, and the abstract structures that link them.  “Logical gaps are to be found at the heart of many hybrids,” Grosholz explains, but imaginative analogies inspire the kind of revision and invention that promotes the growth of mathematical knowledge.

I was always impressed by the fact that these intuitive leaps that Leibniz took, while prompting subsequent generations to feel the need to bring acceptable rigor to the notions, were nonetheless substantiated.  Grosholz lends some important detail to the picture Richard Courant paints of 17th century pioneers of mathematics in his classic text, What is Mathematics?

In a veritable orgy of intuitive guesswork, of cogent reasoning interwoven with nonsensical mysticism, with a blind confidence in the superhuman power of formal procedure, they conquered a mathematical world of immense riches.

This talk of hybrids reminded me of the interdisciplinarity that Virginia Chaitin writes about.  I wrote this in an earlier post about one of her papers:

What she proposes is not the kind of interdisciplinary work that we’re accustomed to, where the results of different research efforts are shared or where studies are designed with more than one kind of question in mind. The kind of interdisciplinary work that Chaitin is describing, involves adopting 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. The results, as she says, are the “migrations of entire conceptual neighborhoods that create a new vocabulary.”

In her own words:

…an epistemically fertile interdisciplinary area of study is one in which the original frameworks, research methods and epistemic goals of individual disciplines are combined and recreated yielding novel and unexpected prospects for knowledge and understanding. This is where interdisciplinary research really proves its worth.

Grosholz’s identification of the hybrid is an important insight, and I would argue that it has implications beyond mathematics.  It may be that because the objects of mathematics are so clean, or unambiguous, the value of the hybrid is more easily observed.   But my hunch is that productive analogies likely belong to the stuff of life itself.

Information and questions of consciousness

I have been particularly concentrated on whether mathematics can tell us something about the nature of thought, something that we have not yet understood about what thought is made from, how it happens, how it is connected to everything else in the universe.  These questions inevitably point me in the direction of research in cognitive science, neuroscience, philosophical debates about the viability of the objectivity on which science relies, and discussions of what we even mean by ‘knowledge.’  Mathematics shows up everywhere, in the abstractions and probabilities involved in how the brain learns, for example, or how the brain constructs what we see, or how the brain navigates the space around us.  One of the avenues I’ve followed has led me through the science of self-organizing systems and the application of information theory to biology in particular, some of which was discussed in a recent post.  In this context, we see biologists exploiting the value of math ideas.  And the modeling that happens in these research efforts doesn’t just predict outcomes.  It often characterizes the action.  The behavior of swarms, flocks, insect colonies, and even cells is mathematical.

It happens in the other direction as well.  Mathematician and computer scientist Gregory Chaitin has approached biology mathematically, not in the sense of modeling behavior, but more in the way of expressing the creativity of evolution using the creativity of mathematics.  Here’s a little piece of a post from about six years ago:

Chaitin believes that Gödel and Turing (in his 1936 paper) opened the door to a provocative connection between mathematics and biology, between life and software. I’ve looked at how Turing was inspired by biology in two of my other posts.   They can be found here and here.

But Chaitin is working to understand it with a new branch of mathematics called Metabiology.  I very much enjoyed hearing him describe the history of the ideas that inspired him in one of his talks:  Life as Evolving Software in which he says:

“After we invented software we could see that we were surrounded by software.           DNA is a universal programming language and biology can be thought of as software archeology – looking at very old, very complicated software.”

Chaitin is also one of the mathematicians who developed what is known as algorithmic information theory.  And I recently happened upon a paper from Giulio Ruffini at Starlab Barcelona, with the title An Algorithmic Information Theory of Consciousness.  This paper was published near the end of 2017.  Ruffini’s research is motivated, to some extent by the value of being able to provide a metric of conscious states.  But the course he’s chosen is described in the abstract:

In earlier work, we argued that the experience we call reality is a mental construct derived from information compression. Here we show that algorithmic information theory provides a natural framework to study and quantify consciousness from neurophysiological or neuroimaging data, given the premise that the primary role of the brain is information processing.

Ruffini argues that characterizing consciousness is “a profound scientific problem,” and progress in this area will have important practical implications with respect to any one of a number of disorders of consciousness.  While the paper is mostly aimed at justifying the fit of algorithmic information theory (which he refers to as AIT) to this endeavor, one can also see some of the deeper philosophical  convictions that motivate his approach.   He says the following, for example, in his introduction:

We begin from a definition of cognition in the context of AIT and posit that brains strive to model their input/output fluxes of information with simplicity as a fundamental driving principle (Ruffini 2007,2009).  Furthermore, we argue that brains, agents, and cognitive systems can be identified with special patterns embedded in mathematical structures enabling computation and compression.

But I found the conviction that seems to be driving his perspective clearly laid out in his 2007 paper Information, complexity, brains and reality (Kolmogorov Manifesto).  There he says that information theory gives us the conceptual framework we need to comprehend how brains and the universe are related. That seems like the really big picture.  He also says:

I argue that what we call the universe is an interpreted abstraction—a mental construct—based on the observed coherence between multiple sensory input streams and our own interactions (output streams).  Hence, the notion of Universe is itself a model. Rules or regularities are the building blocks of what we call Reality—an emerging concept. I highlight that physical concepts such as “mass”, “spacetime” or mathematical notions such as “set” and “number” are models (rules) to simplify the sensory information stream, typically in the form of invariants. The notion of repetition is in this context a fundamental modelling building block.

Compression is one of the key ideas. Relations that are expressed in equations, or events that are captured by programs have been compressed, and the simplification is productive. The Kolmogorov complexity of a data set, in algorithmic information theory, is defined as the length of the shortest program able to generate it.  Experience is a consequence of the brain’s compression (and hence simplification) of an ongoing flood of sensory data.  And so one of Ruffini’s ideas is that science is what brains do.  And this, he says, is to be taken as a definition of science.  Here are a few of the ideas his paper means to address, some more provocative than others:

Reality is a byproduct of information processing.

Mathematics is the only really fundamental discipline and its currency is information.

The nervous system is an information processing tool.  Therefore, information science is crucial to understand how brains work.

The brain compression efforts provide the substrate for reality (the brain is the “reality machine”).

The brain is a pattern lock loop machine.  It discovers and locks into useful regularities in the incoming information flux, and based on those it constructs a model for reality.

…the concept of repetition is a fundamental block in “compression science.”  This concept is rather important and leads brains naturally to the notion of number and counting (and probably mathematics itself)

Compressive processes are probably very basic mechanisms in nervous systems…Counting is just the next step after noticing repetition.  And with the concept of repetition also comes the notion of multiplication and primality.  More fundamentally, repetition is the core of algorithmics.

This sketchy survey of the paper does not do it justice.  But I bring it to your attention as yet another indication that the blend of information theory and biology is running deep.


Probabilities and hallucinations

I’ve written before about how probabilities are used to understand human perception, understanding, and learning.  Joshua Tenenbaum uses probabilistic inferencing to account for how we come to learn concepts, acquire language, and understand the world around us quickly, and with very little information (How to Grow a Mind).   Optical illusions are created by statistical judgments that the brain seems to be carrying out (what we see is the most likely interpretation of sensory data).  In cognitive science, probabilistic programs are used more and more to model cognitive processes.  These programs contain the causal structure of some chosen range of possible events, and use probabilities to address the things that are not known. They run like simulations, providing a way to predict events when run forward, and a way to analyze them when run backward. Many of the models have strong predictive value when applied to real world situations. One such experiment, designed to determine how preschoolers will generalize an event, strongly supports the idea that 15-month old children used an intuitive sense of the effects of sampling to make a generalization in their immediate experience (or to not make one).

Anil Seth, at the University of Sussex, proposes that this nontrivial mathematical idea is responsible, not only for how we perceive and learn about the world around us, but also how we perceive ourselves. His research begins with the observation that the body creates what we see by organizing sensory signals and prior experience with probabilities. For Seth, we don’t so much perceive the world as generate it which, he notes, is consistent with a Helmholtzian view of things (see last month’s post).  And this generation of a world that we experience is not just the organization of signal coming from the outside. The brain is also addressing signal from the inside – like blood pressure or how the heart and internal organs are doing – as much, if not more, than the outside. This side of the brain’s attention concerns control and regulation. Seth argues that the brain is consistently engaged in error-reducing predictive processing, to keep us alive.

With the continuous flow of signal from outside the body and within the body, the brain makes its best guess about what’s happening.  Seth calls our shared experience a controlled hallucination. What we typically call a hallucination occurs when, for whatever reason, the brain pays less attention to signals coming from outside the body and runs more purely on expectations.

For Seth, conscious experience, and our multilayered experiences of selfhood, are also constructed from the organization of sensory data and internal states, and characterized by the most likely meaning of that data. Even our sense of owning our own bodies is a consequence of the brain’s predictions about the data it receives. Experimental evidence supports this claim. The rubber hand experiment, for example, is one in which an individual is asked to focus on a fake hand while their real hand is kept out of sight. The fake hand will begin to feel like part of the participant’s body when tactile stimuli are arranged to suggest this. In another experiment, the electronic image of a hand is perceived as belonging to the body when it is programmed to flash gently red in time with that individual’s heartbeat.

The brain’s probabilistic judgments are not easily unraveled. They are based on a complex set of physiological processes that include the body’s sense of its own position and its own motion, (produced by stimuli that arise from within the body), the body’s sense of its own internal state, as well as what it receives from the five senses that we recognize. What we perceive is some reconciliation of integrated signals and expectation. And when expectations are given priority over incoming data, one will perceive what is not there.

In this paradigm, mathematical processes are found living. They are a feature of our biology. But another important implication of this work is that, with all of these error-reducing guesses that the brain is making (using probabilities and past experience), we could be mistaken about anything – about what’s out there as well as about ourselves. In the history of science, mathematics has consistently corrected mistakes in our perception of reality – from the earth’s position in the solar system, to the calculus of classical mechanics, to the probability-driven theories of quantum mechanics. So it seems to be addressing our view of things from both ends.  As pure structure, related to fundamental life processes, mathematics may also be uniquely capable of clarifying what we can understand about ourselves or, even more importantly, what we can understand about our relationship to what we believe is our reality. Insights gained from a cognitive science perspective must, inevitably, connect to cosmological questions like Wheeler’s participatory universe, or the Qbism view of the wave function described here.  The promise of a genuinely fresh approach to longstanding philosophical issues, like the viability of an objective point of view, intrigues me in these discussions, and mathematics plays a consistently significant role in substantial paradigm shifts like this one.

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.


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.