I recently considered the role that mathematics plays in bringing meaning, or perhaps even story, to our experience. Mathematics is often used to reveal the structure that can be found in large sets of data, or in any number of physical things that change over time, or in the properties of the abstractions themselves. Mathematics, then, sort of tells us what we’re looking at, within what would otherwise be an unwieldy or even meaningless data. In my last post, I took note of the way that mathematics was used to communicate meaning in the structure of cathedral walls – in particular Lincoln Cathedral whose construction began at the end of the 11th century and continued through the Middle Ages. There I shared the suggestion that the geometry of the Cathedral was used to convey, or even to instruct on, the fundamental nature of our being. It happens now that another New Scientist article, Shakespeare: poet, playwright, scientist broadened the issue for me.
The article was written by broadcaster and author Dan Falk, whose most recent book is The Science of Shakespeare. In the article, Falk considers, specifically, evidence in the play’s words that Shakespeare made use of astronomical observations, and explores how he may have known about them.
Galileo’s telescopic observations came toward the end of Shakespeare’s career, but Cymbeline, first performed in 1611, offers tantalizing hints that he was aware of the findings. In the play’s final act, the hero, Posthumus, falls into a dream-like state, and the ghosts of four family members appear and move around him in a circle. The ghosts cry out for Jupiter, the Roman god. On hearing their pleas, he descends onto the stage. So we have Jupiter and four ghosts moving in a circle.
But the book also addresses the emergence of the scientific world view more generally and, in doing so, takes note of the relationship between science and religion. I recently saw a group of students who had gathered on the University of Texas, Dallas campus with posters designed to provoke a debate about truth, science and religion. When I saw them, it occurred to me that mathematics never comes up in these debates, despite the fact that it is so intimately tied to ancient, medieval and modern world views. Reading the history of the relationship between science and religion, as Falk surveys it, encouraged me to pursue my own thoughts about whether an updated philosophy of mathematics might contribute to the science vs. religion debate. Early in his book, Falk makes the following observations:
…the rediscovery of classical texts, via Arabic translations, triggered a new wave of learning across Europe. Those works included the writings of Aristotle and Ptolemy…,as well as the geometry of Euclid, the medical writings of Galen, and much more…This wave of learning was closely linked to the activities of the Roman Catholic Church. The best medieval schools had been those associated with the monasteries and the great cathedrals. By the late Middle Ages these had also become centers of what we would now call science…There were also the universities, the earliest having been founded around 1200; these, too, functioned largely as religious institutions. The highest degree offered was in theology – though to obtain it, the student also had to master mathematics, logic, and natural philosophy.
The close connection between science and faith may seem strange to the modern reader, living at a time when Western society, and Western science in particular, has become a secular endeavor….The evolving relationship between science and religion is a large and complex subject…For one thing, religion was simply part of the fabric of society; all of the key figures of the Scientific Revolution were men of faith of one kind or another.
…As historian Paul Kocher puts it, early modern science “was more often cited as proving God’s existence than disproving it…And as Principe writes, the study of nature was seen as “an inherently religious activity…But the link between science and faith, as Principe stresses, is deeper than this. For the thinkers of early modern Europe, he writes, “the doctrines of Christianity were not personal choices. They had the status of natural or historical facts.”
Mathematics inspired many of the thoughts of an emerging scientific world-view, but it is also closely associated with the more mystical side of things. Kepler, as Falk points out, was obsessed with numerology, like the ancient Pythagoreans. His work relies on observations of the physical as well as the mystical. Quoting Allen Debus, Falk tells us:
His subtle mixture of mathematics and mysticism is “far removed from modern science, but it formed an essential ingredient of its birth.”
Mathematics is often at the beginning of things. The considerations of Shakespeare’s Hamlet explored in Falk’s article, and in his book, should remind us of the complexity of human thought. Relationships among art, science, and religion are inherent. They are all narratives that grow out of the same ground. Their coexistence, it would seem, is just a fact of our existence, and one of the keys to a deeper understanding of who we are.
If mathematics is, as I tend to see it, the mind itself building structure with the elements of thought, then mathematics’ development is like the development of another sense. It is, after all, the structure we give to any sensory data that creates a meaningful thing perceived. And so it is with mathematics. Mathematics, like story and name, brings meaning, not mechanics, to our experience. It increasingly links what we see to what we think we know, as we try to reconcile our immediate experience with what’s ‘out there.’ The evolution of mathematical concepts allows us to probe deeper and deeper into the universe, as well as into our own nature. And this is the subject of both religion and science. So why is mathematics so consistently absent from our debates?
In their March 22 issue, New Scientist reported on the recent detection of gravitational waves that are predicted by the inflationary theory in physics. This observation could help reveal details of what the cosmos was like “in the first slivers of a second” following the big bang. It supports the theory that implies the existence of an ever-expanding multiverse. There’s also a nice write up in the Guardian, which is free.
Lisa Grossman, the author of the report, followed up with a short piece (Medieval text about light held hints of a multiverse) describing a recent paper which explores the mathematical side of a 13th-century treatise on light written by Robert Grosseteste. This paper is one of the fruits of an interdisciplinary research project at Durham University, UK, which has been given the name The Ordered Universe. It’s core team of investigators includes Giles Gasper, from Durham’s Department of History, Tom McLeish from physics, Cecilia Panti from the University of Rome’s Department of Philosophy, and Hannah Smithson from the Department of Psychology at Pembroke College, University of Oxford, UK. This group along with Richard G. Bower, Brian Tanner and Neil Lewis, co-authored the paper in which Grosseteste’s treatise On Light (De luce) is reformulated in terms of modern mathematics. The authors conclude that Grosseteste’s ideas contain the prospect of the multiverse theories now being considered by cosmologists.
This kind of observation may help reveal something about the kinship between mathematics and thought itself. Grosseteste brought a conceptual order to non-mathematical thoughts. His own thoughts follow the patterns that he finds in his experience, in the thoughts of others, as well as in the very conceptual structures he considers. He follows the implications of the ideas he puts forward and then creatively interprets these implications.
Yet this non-mathematical order can accommodate the superpositioning of modern mathematics, out of which emerges something remarkably similar to contemporary theories in cosmology.
The abstract of the paper reads as follows:
In his treatise on light, written in about 1225, Robert Grosseteste describes a cosmological model in which the Universe is created in a big-bang like explosion and subsequent condensation. He postulates that the fundamental coupling of light and matter gives rises to the material body of the entire cosmos. Expansion is arrested when matter reaches a minimum density and subsequent emission of light from the outer region leads to compression and rarefaction of the inner bodily mass so as to create nine celestial spheres, with an imperfect residual core. In this paper we reformulate the Latin description in terms of a modern mathematical model. The equations which describe the coupling of light and matter are solved numerically, subject to initial conditions and critical criteria consistent with the text. Formation of a universe with a non-infinite number of perfected spheres is extremely sensitive to the initial conditions, the intensity of the light and the transparency of these spheres. In this “medieval multiverse”, only a small range of opacity and initial density profiles lead to a stable universe with nine perfected spheres. As in current cosmological thinking, the existence of Grosseteste’s universe relies on a very special combination of fundamental parameters.
It is this “very special combination of fundamental parameters,” where the nine-sphere universe depends on key initial conditions, that opens the door to a multiverse. Page two includes a careful description of the paper’s intent:
While it is crucial to avoid superposing a modern world view into Grosseteste’s thought, throughout his work there pervades an interest in the nature of the created world, the existence of order within it, the mechanisms whereby it is sustained and a search for unity of explanation. These ideas are common in medieval thinking; nonetheless the originality of Grosseteste was to think about unity, order and causal explanation of natural phenomena as being due to light, its properties and the mechanism by which we perceive it. …We are not trying to “correct” Grosseteste’s thinking in the light of modern physics, nor are we claiming Grosseteste’s ideas as a precedent for modern cosmological thinking. Rather, we are making a translation, not just from Latin into English but from the new critical Latin edition  and English translation  of his De luce into mathematical language. We aim to write down the equations, as he might have done had he access to modern mathematical and computational techniques, solve the equations numerically and explore the solutions. There are benefits here from both an historical and a scientific perspective. The application of mathematics and computation generate, as we shall see, a closer and more comprehensive examination of a medieval scientific text and the mind behind it. However, there are scientific benefits as well, as the medieval cosmos constitutes a quite novel arena to compute radiation/matter interactions and dynamics, and in which to discover new physical structure.
Grosseteste’s work builds on Aristotle’s idea that the earth is embedded in a series of nine concentric spheres that are the universe. Particularly striking, however, is Grosseteste’s idea that light is the “first corporeal form and that it multiplies itself infinitely.” He argued that light (luxe) expanded instantaneously from a point. For Grosseteste, this first corporeal form has no dimension and, since form and matter are inseparable, neither does matter. Only by its expansion into all directions does light introduce three dimensions into matter.
In the beginning of time, light extended matter, drawing it out along with itself into a sphere the size of the material universe.
Light drags matter outwards and so, Grosseteste concludes, the density of matter must decrease as its radius increases.
But finding Grosseteste’s cosmology in the architecture of Lincoln Cathedral makes things even more interesting. John Shannon Hendrix at Roger Williams University does just that in his 2010 paper, “The Geometries of Robert Grosseteste and the Architecture of Lincoln Cathedral.”
…the geometries were similar enough to suggest that a cultural concept of a geometrical substructure of matter could be translated into the architectural forms of the cathedrals as catechisms of the structure of matter and being. The architecture of the cathedrals was intended to edify the viewer as to the underlying nature of being and as to the relation between the human intellect and nature and God.
…Geometries used by Grosseteste to describe the diffusion and rarefaction of light in the formation of matter can be compared to the peculiar geometries of the vaulting of Lincoln Cathedral, in the particular lines and line segments of the vaulting. Volumes formed by the reflection and refraction of light, as described by Grosseteste, can be compared to the volumes of the vaulting, in particular the concave and conical shapes. The lux, or spiritual light, and the lumen, or physical light, can be applied to the light in the cathedral, as shining through the stained glass windows and illuminating the geometries. (emphasis added)
The Stanford Encyclopedia of Philosophy makes the following statements about Grosseteste:
Under Augustine’s influence Grosseteste subscribes to an illuminationist account of human knowledge, according to which human knowledge is understood by analogy to bodily vision: as a body can only be seen if light is shed on it and the eyes, so something can only be known if a spiritual light is shed on it and the mind’s eye. Grosseteste presents versions of such an account in his treatise On Truth and in his commentary on the Posterior Analytics.
…But as we can only see a body as colored if an external light is shed on its color, rendering it actually visible to us, so in order for us to see a created thing as true, an external light must be shed on its truth, rendering it actually visible to us. (emphasis added)
What we have here are a series of overlapping metaphors, if you like, each trying to capture (sensorily and intellectually) our own human impression of our existence and our origin. It may be said that mathematics brings us most precisely (in the intellectual sense) to the beginning of things, but mathematics, in all of them, is used to build and to communicate meaning, bringing meaningful structure to remote light signals, as well as cathedral walls. That it can speak to us about ourselves in so diverse a manner again indicates its fundamental nature in human experience.
A Scientific American article brought mathematical knitting to my attention once again, and within the article was a link to Bridges, an organization which oversees the annual Bridges conference that explores the connections between art and mathematics. Following the link to their 2013 Conference, I found their Short Movie Festival. I’ve watched a number of these short films whose actions explore things like topological objects, fractals, geometric symmetries, and groups. These ideas are presented in visual works, movement performances, as well as in nature. I thoroughly enjoyed all of them and I found myself asking the question, “So where does mathematics live, really?” For example, associated with the short movie Dancing Braids by Ester Dalvit is the following note:
Braids can be described as configuration space of points in a disc. These can be visualized as dances: the positions of each dancer are translated into a strand of the braid, the time into a spatial dimension.
This movie is a small part of a long video about braid theory which is available here.
Or with Susan Gerofsky’s film, The Geometry of Longsword Locks, is this:
In traditional English longsword dancing, a team of dancers makes intricate moves while joined together by their wooden or metal ‘swords’. An impressive element of the dance is the variety of traditional geometric, symmetrical sword locks (often stars) created through the movements of all the dancers. The film showcases a longsword dance and the locks created by the physical algorithms of the conjoined dancers’ movement. After showing the dance, questions are offered to spark mathematical explorations by secondary or post-secondary students. These questions include topological and geometric ideas about crossings, angles and edges, and logic-related questions about categorizing lock types and discovering whether new locks could be created through analysis of the physical algorithms that create them. Slow-motion and repeated views help learners explore this rich source of geometry.
On the Simons Foundation website is yet another short video on Change Ringing.
The art or “exercise” of change ringing is a kind of mathematical team sport dating from the 1600s. It originated in England but now is found all over the world. A band of ringers plays long sequences of permutations on a set of peal bells. Understanding the patterns so they can be played quickly from memory is an exact mental exercise which takes months for ringers to perfect. Composers of new sequences must understand the combinatorics of permutations, the physical constraints of heavy bells, and the long history of the art and its specialized vocabulary. Change ringing is a little-known but surprisingly rich and beautiful acoustical application of mathematics.
According to The North American Guild of Change Ringers,
the earliest record we have of these is from 1668:Tintinnalogia: or, The Art of Ringing. Wherein Is laid down plain and easie Rules for Ringing all sorts of Plain Changes. Together with Directions for Pricking and Ringing all Cross Peals; with a full Discovery of the Mystery and Grounds of each Peal.
Perhaps we can ignore the effect of the subject tabs we learned to put in our notebooks when we were young and ask some new questions. Do these visual and musical experiences represent mathematical concepts or are mathematical concepts actually exploring the elements of these visual and musical experiences? I lean in the direction of the latter. In fact, I would argue that one of the major functions of the brain is to integrate experience. The dances shown in two of the short films are, in some sense, an impulsive integration of the things we hear, see and hold, that become shapes within the inherent unity of our experience. It can be said that mathematics ‘picks up’ on this impulse, and further explores that unity by investigating the paths that are born of these more impulsive harmonies. Mathematics is then distinguished by its symbolic representation of the flow of patterns created by our living – by the visual, and audio structures that the senses build, as well as the cognitive structures that develop with them. Braids and knots are two of the oldest human impulses to create new experience, and they are two of newest objects investigated by mathematics, which then further integrates them into what we know of number and quantity and symmetry.
There is one more thing, not so much related to the theme of this post, but worth a look. One of the short movies in the Bridges short movie festival is a poetic approach to the words real and complex that I think is really nicely done. You can go to it directly here.
Back in September, 1992 Semir Zeki wrote an article for what was then a special issue of Scientific American called Mind and Brain. In it he described what was known about how the brain produces visual images. I have referred back to the article many times because it highlights the philosophical implications of our current grasp of these processes. Right below the title of the article was this remark:
In analyzing the distinct attributes of images, the brain invents a visual world.
Near the end he makes an important observation:
The past two decades have brought neurologists many marvelous discoveries about the visual brain. Moreover, they have led to a powerful conceptual change in our view of what the visual brain does and how it accomplishes its functions. It is no longer possible to divide the process of seeing from that of understanding, as neurologists once imagined, nor is it possible to separate the acquisition of visual knowledge from consciousness. Indeed, consciousness is a property of the complex neural apparatus that the brain has developed to acquire knowledge.
Zeki’s investigation of the visual brain has lead to a significant amount of work on the neurobiology of aesthetics. He heads the Institute of Neuroaesthetics at University College London. VisLab, The Artificial Vision and Intelligent Systems Laboratory at the University of Parma, Italy, has contributed to the institute’s work. Within an introduction to the institute’s purpose, and with respect to Vislab in particular, there is the following statement:
Over the past few years Vislab has contributed to neuroesthetics by exploring visual art in relation to the known physiology of the visual brain.
Underlying the approach are three suppositions:
• that all visual art must obey the laws of the visual brain, whether in conception or in execution or in appreciation;
• that visual art has an overall function which is an extension of the function of the visual brain, to acquire knowledge;
• that artists are, in a sense, neurologists who study the capacities of the visual brain with techniques that are unique to them.
Very recently, Zeki co-authored a paper on The experience of mathematical beauty and its neural correlates. Neuroscientist, John Paul Romaya; physicist, Dionigi M. T. Benincasa; and mathematician, Michael Atiyah, were his co-authors. The paper was published in the journal Frontiers on February 13. Their study, was aimed at determining whether the beauty experienced in mathematics correlates with activity in the same part of the emotional brain (referred to as field A1 of the medial orbito-frontal cortex or mOFC) as the beauty derived from sensory or perceptually-based sources like visual art and music. Their results showed that mathematical beauty was correlated with activity in this part of the emotional brain, which raises some interesting questions related what our experience of beauty is all about.
Unlike studies that looked at the neurobiology of musical or visual beauty, this study required the recruitment of individuals with a fairly advanced knowledge of mathematics. And so, while it may be difficult to sort out, this effort, the author’s suggest,
…carried with it the promise of addressing a broader issue with implications for future studies of the neurobiology of beauty, namely the extent to which the experience of beauty is bound to that of “understanding.”
The study included 12 non-mathematical subjects, but the majority of these individuals indicated that they didn’t understand the equations and that they didn’t have an emotional response to an equation they may have found beautiful (despite the fact that some did rate particular equations as beautiful). Researchers were able to parse the components of the non-mathematicians’ judgment to some extent. Finding more intense activity in the brain’s visual areas for these subjects, confirmed their hunch that the beauty-rating from the non-mathematical participants was a judgment about the formal qualities of the equations – the forms displayed, their symmetries, etc.
The paper becomes even more interesting when the authors consider the implications of their work:
The experience of beauty derived from mathematical formulations represents the most extreme case of the experience of beauty that is dependent on learning and culture. The fact that the experience of mathematical beauty, like the experience of musical and visual beauty, correlates with activity in A1 of mOFC suggests that there is, neurobiologically, an abstract quality to beauty that is independent of culture and learning. But that there was an imperfect correlation between understanding and the experience of beauty and that activity in the mOFC cannot be accounted for by understanding but by the experience of beauty alone, raises issues of profound interest for the future. It leads to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved. (emphasis added)
And then a quote from a talk given by Paul Dirac in 1939 (one of the subjects of an earlier post of mine), where Dirac advices physicists to look first at promising mathematical ideas, and to consider beauty over simplicity.
There is no logical reason why the (method of mathematical reasoning should make progress in the study of natural phenomena) but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme. . . What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty… The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. . . We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty.
What I find very encouraging is that this paper suggests, in yet another way, a coupling of the body with its world that mathematics may yet have a hand in helping to reveal.
The Platonic tradition would emphasize that mathematical formulations are experienced as beautiful because they give insights into the fundamental structure of the universe (see Breitenbach, 2013). For Immanuel Kant, by contrast, the aesthetic experience is as well grounded in our own nature because, for him, “Aesthetic judgments may thus be regarded as expressions of our feeling that something makes sense to us” (Breitenbach, 2013). We believe that what “makes sense” to us is grounded in the workings of our brain, which has evolved within our physical environment…Hence the work we report here, as well as our previous work, highlights further the extent to which even future mathematical formulations may, by being based on beauty, reveal something about our brain on the one hand, and about the extent to which our brain organization reveals something about our universe on the other.
I was intrigued by a paper that came to my attention in the December 2013 issue of Philosophical Studies by Anna Farennikova in which she argues that we ‘see’ absence. In other words, seeing that something is not there is as much a product of our visual system as seeing an object. The example with which she begins goes like this. “If someone steals your laptop at a cafe, you may see its absence from your table.” Can you “see” something that is not present? Farennikova rejects the idea that the visual information is simply the table (without the laptop) and that the absence of the laptop is quickly deduced. She argues, instead, that our visual system includes mechanisms for ‘seeing’ the absence of something, making the case that “in addition to representing objects, perception represents absences of objects.” You might be tempted to say “what difference does it make if I ‘see’ the absence of something or ‘judge’ the absence of something. But Farennikova explains the difference that it makes:
The phenomenon of seeing absence can thus serve as an adequacy-test for a theory of perceptual content. If experiences of absence are possible, then we have another reason (following Siegel) to reject the view that perceptual content is restricted to colors and shapes.
This is a question that addresses what it means to perceive. And this is exactly why it interests me. The argument she builds is one that necessarily considers the variation in sensory experience, particularly in visual experience. There are subtleties in the distinction between sensory experience and higher level cognitive experience. Observing these nuances inevitably leads to a careful evaluation of what it means ‘to perceive,’ which is important to some of the arguments I’ve made about the nature of mathematics.
Farennikova does a fairly thorough job of anticipating her critics. And she is careful to distinguish the phenomenon that she is addressing from other experiences where there is a ‘failure to see.’ She draws attention to the fact that “many experiences of absence feel instantaneous and lacking in conscious effort.” She also points to the strong adaptive advantage of seeing absences.
To survive, we need to be reliably and efficiently informed not only about “what is present in the world, and where it is” (Marr 1982), but also about what is absent from the world and where it is absent. This reliability may require automaticity, which is a function of blocking interference from beliefs and higher cognitive states. If these reasons are correct, then the capacity to sensorily respond to the absence of things should be as primitive and fundamental to humans as the capacity to sensorily respond to the presence of things.
Farennikova’s argument relies on specifying the mechanism involved in experiences of absence, and showing that this mechanism is visual as opposed to cognitive. The model she proposes is a matching operation, where templates that have developed through experience in the visual system are matched with the sensory input of any given moment. These templates of absent objects are not images. They preserve some of the visual attributes of the object but they also hold more abstract information about how the structure of the object is organized. Templates are generated in sensory memory and, she explains, exist at a subpersonal level. They are not necessarily the same as conscious imagery and are not dissimilar to the processing considered commonplace in ordinary vision. Since all of these components are visual, it is reasonable to regard the entire process as visual.
Farennikova then appeals to the rich content view of seeing:
Theories of seeing have been tailored to the perception of material objects, so it is no surprise that absences fail to satisfy their criteria. But what justifies the assignment of genuine seeing only to material objects?
In some visual experiences, some properties other than spatial properties, color, shape, motion, and illumination are represented” (Siegel 2010 The contents of visual experience. Oxford University Press)
I also found a paper that takes issue with Farennikova by Jean-‐Rémy Martin, Université Paris and Jérôme Dokic, Ecole des Hautes Etudes en Sciences Sociales.
While agreeing with Farennikova that absence experiences are not reducible to high-level cognitive states such as beliefs, we reject the Perceptual View. Instead, we claim that absence experiences are neither strictly perceptual nor strictly cognitive. In particular, we propose that these experiences belong to the category of metacognitive (specifically metaperceptual) feelings, which reflect a specific kind of affective experience caused by subpersonal monitoring of (perceptual) processes.
Absence experiences for these authors are metaperceptual feelings of suprise. They argue that “mismatches at the narrow level of templates” are not enough to produce experiences of absence. “Rather, the adaptive function of the experience of incongruity suggest that it must be driven by whole expectations (with templates as proper parts).”
These things may all seem like tedious distinctions, but within this discussion there are actually some intriguing questions, like: When is the work of sensory processing over? When do our conscious minds appear to be at the helm? Is there anything in the middle? I’m inclined to say that even at the advanced level of cognition that we call mathematics, sensory processes are still at work. Not just in the reading of notation or the analyzing of images, but in the content of the mathematics itself, where templates, matches and mismatches are likely moving our minds eye to find out what’s there and what it means. Perhaps there is no cut off between sensory and higher-level cognitive processing. Rather there is some continuum of seeing/understanding whose depths are explored abstractly in mathematics and, in a more immediate way, by language.
Note: Farennilova has put together a series of images related to her research on her website : Seeing Absence
This is something of a follow-up to my last post. I checked out a series of links related to Max Tegmark in the last few days, having heard about the release of his first book Our Mathematical Universe. But I was also motivated by having observed that the latest conference organized by the Foundational Questions Institute (for which Tegmark is one of the directors) included prominent neuroscientists, Christof Koch and Giulio Tononi. This is not the first time a FQXi conference has included neuroscientists among their list of speakers. There are a series of threads that one can follow through Tegmark’s and Tononi’s work, but I would like to make a particular observation. Tegmark’s thesis in Our Mathematical Universe, and Tononi’s strategy in his 2008 paper on ‘Consciousness as Integrated Information,’ each rely on the significance of pure ‘relations,’ in how we analyze our experience as well as in how our experience is produced.
Tegmark has been arguing that the universe itself is a mathematical object or structure. His book is a full treatment of this idea. One of the keys to his defense of this idea is the claim that, as theories in physics have developed, their content has become more and more purely relational. In a 2007 paper that preceded the recent book, Tegmark explains that all of the physical theories that have been produced thus far have two components: mathematical equations and what he calls “baggage,” or the words that we give to the relations when we describe them.
However, could it ever be possible to give a description of the external reality involving no baggage? If so, our description of entities in the external reality and relations between them would have to be completely abstract, forcing any words or other symbols used to denote them to be mere labels with no preconceived meanings whatsoever.
A mathematical structure is precisely this: abstract entities with relations between them.
He then says later:
In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics.
With a more recent paper (Jan. 2014), Tegmark takes on the nature of consciousness. In Consciousness as a State of Matter, he brings principles of physics into a discussion of consciousness. He proposes the possibility that consciousness can be understood as a state of matter, like the states of matter we call a liquid, a solid and a gas and then begins an analysis of the properties that such a state of matter would have. When enormous numbers of particles are brought together, he explains, new and interesting emergent phenomena begin to happen. And while there are a large number of kinds of gasses, there is an independent substrate that they all share. These kinds of ideas can be brought to an analysis of the states of matter that define consciousness as well. One of the properties of memory, for example, is that it has many long-lived stable states. It also has dynamic properties. So the question becomes, can one take the ideas in neuroscience and use them to say something interesting about the physical world? Why do we perceive ourselves, for example, as living in a 3-dimensional space with a hierarchy of objects? How do we get there from the fundamental properties of matter described by modern physics?
Using some of the mathematics that describes physical systems, Tegmark tries to find the way that our experience would emerge from (he actually says “pop out”) of the mathematics. He calls it the ‘physics from scratch problem.’ Tegmark’s paper means to extend Tononi’s work on consciousness to more general physical systems by using information theory and Tononi’s idea of integrated information. He is convinced that the problems of neuroscience and the problems of physics are very strongly linked.
Can a deeper understanding of consciousness breathe new life into the century-old quest to understand the emergence of a classical world from quantum mechanics, and can it even help explain how two Hermitean matrices H and ρ lead to the subjective emergence of time? The quests to better understand the internal reality our mind and the external reality of our universe will hopefully assist one another.
Tononi’s paper finds experience to be the mathematical shape given to integrated information. Information is defined as the reduction of uncertainty. And it is the discrimination among alternatives that generates information. Tononi proposes a way to characterize experience with a geometry that describes informational relationships. The integration of information produces a ‘shape’ in what he calls qualia space, and a particular shape is a particular experience. The ‘space’ is defined using a set of axes each labeled with probabilities related to the states of a system in the brain (like visual systems) and the interactions among elements in the system. When a large number of elements and connections are at play, the dimension of the quailia space far exceeds three. For example, four elements with nine connections among them is a simple system, but it produces a 16-dimensional space. About these shapes Tononi writes that they are
often morphing smoothly into another shape as new informational relationships are specified through its mechanisms entering new states. Of course, we cannot dream of visualizing such shapes as qualia diagrams (we have a hard time with shapes generated by three elements). And yet, from a different perspective, we see and hear such shapes all the time, from the inside, as it were, since such shapes are actually the stuff our dreams are made of— indeed the stuff all experience is made of.
And then there’s this bit of poetry in the paper:
If one accepts these premises, a useful way of thinking about consciousness as a fundamental property is as follows. We are by now used to considering the universe as a vast empty space that contains enormous conglomerations of mass, charge, and energy—giant bright entities (where brightness reflects energy or mass) from planets to stars to galaxies. In this view (that is, in terms of mass, charge, or energy), each of us constitutes an extremely small, dim portion of what exists—indeed, hardly more than a speck of dust.
However, if consciousness (i.e., integrated information) exists as a fundamental property, an equally valid view of the universe is this: a vast empty space that contains mostly nothing, and occasionally just specks of integrated information —mere dust, indeed—even there where the mass-charge–energy perspective reveals huge conglomerates. On the other hand, one small corner of the known universe contains a remarkable concentration of extremely bright entities (where brightness reflects high levels of integrated information), orders of magnitude brighter than anything around them. Each bright “star” is the main complex of an individual human being (and most likely, of individual animals). I argue that such a view is at least as valid as that of a universe dominated by mass, charge, and energy.
In a talk given by Tegmark for the “Philosophy of Cosmology” project, he makes the claim that perhaps physical existence and mathematical existence are the same. The view of mathematics proposed by Tegmark and supported by Tononi seem to reverse the embodiment ideas first presented by George Lakoff and Raphael Nunez in their book, Where Mathematics Comes From. The idea analysis in the Lakoff/Nunez book rests on the claim that mathematical concepts develop, through effective metaphors, from fairly simple, fundamental, physical experience. In Tegmark’s world, at least, the mathematics comes first.
I just saw The Guardian’s Science Weekly podcast for November 11, 2013 which included a discussion with mathematician Edward Frenkel about his new book Love & Math: The Heart of Hidden Reality. I then listened to a Huffington Live segment from January 7 where Max Tegmark and Brian Greene talked about the link between mathematics and reality. Tegmark was speaking from the perspective of his new book, Our Mathematical Universe. I’ve only just ordered both books and so I haven’t read them yet. But I would like to say a few things about how each of the authors introduced their ideas. While I find both works encouraging, both bold attempts to reorient the popular view of mathematics, I’m struck by how different they are.
Edward Frenkel began by addressing the need to bring, to a broad audience, some heart-felt appreciation of the beauty in mathematics. The reason no one can see it, he suggests, is the fault of teachers, and not entirely because the ideas are complex. Equally complex ideas like space-time, quantum mechanical behavior,black holes, the Higgs particle, even DNA have found their way into the popular culture. There is no reason that the real subject of mathematics can’t be made similarly accessible for a popular audience. I very much agree.
Frenkel also spent a fair amount of time talking about the Langlands Program, which he referred to as the effort to find a ‘grand unified’ theory of mathematics. In this discussion, he imagined the different branches of mathematics as continents, fully separated land masses. When one finds (as Langland did) a way to translate questions from one area into questions in another, mathematics, Frankel tells us, becomes a teleportation device. While this work has some obvious pragmatic implications, Frenkel uses this idea to point out that the deeper impact of the Langlands Program is in how it reveals the way things (in a more general sense) are connected. Mathematics, he says, tells us about hidden structure that “we still don’t see.” “The more we know about mathematics,” he continues, “the more tools we will have to understand how the world works.” He likens mathematics to an unfinished jigsaw puzzle that’s giving us glimpses of a hidden reality whose final image we don’t know.
Notice he doesn’t say “the more mathematics we know,” but rather, “the more we know about mathematics.” One of the reasons that mathematics seems dry and uninteresting to so many is that it is rarely thought of as an exploration in itself, a search for new meaning and new possibilities or, as Frenkel puts it, for an image you have not seen before.
Frenkel also makes the argument that everything in our world is migrating to the digital. Three-dimensional printers “will be able to print everything out on demand, like a table and spoon, and so on.” The deepest level of physical reality is becoming a digital layer. In this world, he exclaims, “mathematics is going to be king!” And this is because “mathematics’ function is to order information.”
When asked about how the universe could be mathematical, Max Tegmark begins with an appeal to the progress made in modern physics. While the properties of familiar objects were once reduced by science to the properties of atoms, the properties of elementary particles have now been reduced to numbers. We give names to the numbers like spin and charge, but they’re really just numbers. But the way Tegmark speaks about the mathematical universe still feels like he’s giving that mathematical universe the kind of independent, objective reality which I think hampers the effectiveness of seeing things this way. I completely agree that what he proposes leaves the doors to our future understanding wide open. The only limits that exist would be the limits of our creativity and imagination. But because his mathematics still seems to stand outside of us, the question inevitably arises (and did during this interview) about whether consciousness or emotion will ultimately be described mathematically. He and Brian Greene seem optimistic that the answer to that question is yes. Frenkel, on the other hand, has already overlapped mathematics and love in a film he co-created called Rites of Love and Math.
I find Frenkel’s perspective more familiar. I haven’t yet seen the film nor read the book but I will do both shortly. I’m also interested in the conference, organized by the Foundational Questions Institute, and still underway today. It’s a conference on the Physics of Information, and includes a session on Mind, Brain, information and consciousness with neuroscientists Giulio Tononi and Christof Koch.
My views on mathematics share something with both Frenkel and Tegmark, but I’m also influenced by an idea described by Humberto Maturana and Francisco Zarela in their 1987 work, The Tree of Knowledge. A key to their understanding of cognition is their idea of ‘coupling.’ Structural coupling, in its most fundamental sense, is described as “a history of recurrent interactions leading to the structural congruence between two (or more) systems.” It is possible that we are beginning to glimpse something of our own coupling – the coupling of our conscious mind with the world we inhabit (a world that grows with our thoughts). And that may be why we can’t decide which side the math is coming from.
An article in the December issue of Scientific American gave me a new insect behavior to ponder and one that might reveal, in the insect’s biology, a distant cousin to the mathematical idea we call mapping. It seems that there are insects that have a talent for recognizing faces. Their talent has much in common with our own facial recognition abilities, except for the brain that carries it. It was a young graduate student, researching the social lives of a particular kind of wasp, who first noticed that she could distinguish individual wasps by their facial markings. Elizabeth Tibbetts (now an Associate Professor at the University of Michigan) then wondered if the wasps identified each other by these markings and began research directed at answering that question. What she found was that the wasps did respond to changes in facial markings and did use variation in facial patterns for individual recognition. Tibbetts further investigated this aspect of their behavior by trying to train them to learn to differentiate among patterns other than their own facial markings. The wasps quickly learned to accurately select among wasp faces, but had noticeably more difficulty discriminating among the other images. This is taken as strong evidence that wasps have neural systems specialized for wasp face recognition. Although honeybees do not use face recognition mechanisms in their own daily lives, researchers have successfully trained them to discriminate human faces. And further,
…although the bees learned faces slowly as compared with P. fuscatus wasps and humans, they were able to develop some ability to process faces holistically, even though they are not hardwired to do so, as P. fuscatus wasps and humans are. Second, honeybees were able to learn multiple view points of the same face and interpolate based on this information to recognize novel presentations. For example, after a bee learns the front and side view of a particular face, it will be able to correctly choose a picture of the same face rotated 30 degrees, even if it has not previously seen this particular image.
Inherent to facial recognition mechanisms is some idea of spatial arrangement as well as transformation (when presented with multiple views of the same face). So I searched out what has been understood about our own neural systems and found an article that Carl Zimmer wrote for Discover in January 2011. The article describes a model for understanding how our own facial recognition mechanisms work. The idea was first proposed 25 years ago by Tim Valentine and Vicki Bruce, two psychologists at the University of Nottingham, and it has been consistently gaining support. Zimmer reported on the work of cognitive neuroscientist Marlene Behrmann, at Carnegie Mellon University who, with her colleagues, had made some important observations when they set out to compare the brains of individuals who are face-blind to those who are face-sighted.
Valentine and Bruce argued that our brains do not store a photographic image of every face we see. Instead, they carry out a mathematical transformation of each face, encoding it as a point in a multidimensional “face space.” (emphasis added)
On a map of face space, you might imagine the north-south axis being replaced with a small-mouth- to-wide-mouth axis. But instead of three different dimensions, like the space we’re familiar with, face space may have many dimensions, each representing some important feature of the human face. Just as the ancient cosmos was centered on Earth, Valentine and Bruce argued that the facial universe is centered on the perfectly average face. The farther a face is from this average center, the more extreme it becomes.
This is not just a model for describing what happens. It is understood as a way to characterize how the brain can negotiate the infinite possibilities that emerge from our experience.
…it offers an elegant explanation for how we can store so many images of faces in our heads. By reducing a face to a point—creating a compact code for representing an infinite number of faces—our brains need to store only the distance and direction of that point from the center of face space. Face space also sheds light on the fact that we are more likely to correctly identify distinctive faces than typical ones. In the center of face space, there are lots of fairly average faces. Distinctive faces dwell far away from the crowd, in much lonelier neighborhoods.” (emphasis added)
This is exactly how our consciously defined mathematical ideas can operate in our experience. They corral infinite possibilities into well defined concepts whose behavior can be observed and interpreted. In this way it is an arrangement of ideas, that compacts experience and often extends the reach of cognition – of our ability to see and understand our world.
Another kind of learning is discussed in an excerpt from the book Incognito that also appeared in Discover in October of 2011 under the title Your Brain Knows a Lot More Than You Realize. The author of the piece is neuroscientist David Eagleman. Eagleman highlights learning that can only be accomplished with a kind of trial and error training. He describes what has come to be called chick sexing – the process of quickly determining the sex of chicks that have been hatched.
The Japanese invented a method of sexing chicks known as vent sexing, by which experts could rapidly ascertain the sex of one-day-old hatchlings. Beginning in the 1930s, poultry breeders from around the world traveled to the Zen-Nippon Chick Sexing School in Japan to learn the technique.
The mystery was that no one could explain exactly how it was done. It was somehow based on very subtle visual cues, but the professional sexers could not say what those cues were. They would look at the chick’s rear (where the vent is) and simply seem to know the correct bin to throw it in.
And this is how the professionals taught the student sexers. The master would stand over the apprentice and watch. The student would pick up a chick, examine its rear, and toss it into one bin or the other. The master would give feedback: yes or no. After weeks on end of this activity, the student’s brain was trained to a masterful—albeit unconscious—level.
Eagleman also discussed what he called flexible intelligence saying the following:
One of the most impressive features of brains—and especially human brains—is the flexibility to learn almost any kind of task that comes their way… This flexibility of learning accounts for a large part of what we consider human intelligence. While many animals are properly called intelligent, humans distinguish themselves in that they are so flexibly intelligent, fashioning their neural circuits to match the task at hand. It is for this reason that we can colonize every region on the planet, learn the local language we’re born into, and master skills as diverse as playing the violin, high-jumping, and operating space shuttle cockpits.
And a clear reference to mathematics is made in the article’s introduction:
Eagleman’s theory is epitomized by the deathbed confession of the 19th-century mathematician James Clerk Maxwell, who developed fundamental equations unifying electricity and magnetism. Maxwell declared that “something within him” had made the discoveries; he actually had no idea how he’d achieved his great insights. It is easy to take credit after an idea strikes you, but in fact, neurons in your brain secretly perform an enormous amount of work before inspiration hits. The brain, Eagleman argues, runs its show incognito. Or, as Pink Floyd put it, “There’s someone in my head, but it’s not me.
There are a few things going on here. First, I’m intrigued by the presence of what could be called ‘mappings’ in the biology of a creature so different from us. This suggests the possibility that a mathematics-like mechanism is not just the product of mammalian brain processes. I’m also impressed by the similarity between the face space model for face recognition and the conceptual compression that defines mathematical ideas in general. Finally, I’m always interested in the extent to which our sense making and even our mathematical insight happens outside of our awareness.
I’m planning to post something new this week, but I would also like to share the link to my guest blog for Scientific American that was posted last week. The title of the piece is: To What Extent Do We See With Mathematics?
Hope you enjoy it.
Much of the research done in cognitive science is designed to study the development of concepts – internal representations that define the idea-driven nature of modern human experience. And, in our experience, it’s difficult to mend the rift that’s been created between what we call thought and what we call reality. But a number of paths have opened up within various ‘embodied mind’ theses that are intended to correct the mistaken duality. I’ve covered some of them in previous blogs. I’ll refer here, again, to biologist and philosopher Humberto Maturana, a proponent of a closely related idea known as enactivism. In a paper that appeared in Cybernetics and Human Knowing in 2002, Maturana refuted the idea that language is a collection of abstract representations that correspond to concrete things, and made the following observation:
Part of the difficulty in understanding the relation between language and existence rests on the view of language as a domain of representations and abstractions of entities that pertain to a different concrete domain. Yet language is not so, languaging occurs in the concreteness of the doings of the observer in his or her actual living in the praxis of living itself…Nothing exists outside the networks of conversations through which we bring forth all that exists, from ourselves to the cosmos that makes us possible…
Languaging is action. And this is how I have come to see mathematics, as action.
Despite the development of embodiment theories (one of the most well known being the Lafoff/Nunez book Where Mathematics Comes From) debates over the objective reality of mathematics versus the imaginative reality of mathematics are hardly resolved. But the distinction between these alternatives is blurred when mathematics is seen as action. All of this came to mind today when I read about the recently discovered and oldest known sundial.
A tire size stone, found marking a Bronze Age grave, had been carved out to mark time with the movement of the sun. It was determined that the stone could reflect half hour increments. The burial ground in the Ukraine, where the stone was found, dates back to the 12th or 13th century B.C. An article posted on livescience in early October describes the find and some of the details about the stone.
To verify that the stone was a sundial, archeologist Larisa Vodolazhskaya calculated the angles that would have been created by the sun and shadows at that latitude. She confirmed that the carvings on the slabs marked the hours accurately. “They are made for the geographic latitude at which the sundials were found,” she said. And these ancient carvings rely on a sophisticated grasp of geometry.
The circular depressions, placed in an elliptical pattern, are hour marks of an analemmatic sundial; the largest groove on the plate, Vodolazhskaya said, marks where the vertical, shadow-casting gnomon would have been placed at the winter solstice.
Meanwhile, a long carved line transected by a number of parallel grooves in the center of the slab would have acted as a linear scale for a more traditional horizontal sundial, where the hours are marked by a gnomon’s shadow falling along hour lines. In this case, the horizontal sundial actually had two gnomons, Vodolazhskaya said. One gnomon tracked the time in the morning hours and early afternoon, and the second covered from late morning to evening, measuring time in half-hour increments. Ancient sundials with half-hour marks are rare, though one was discovered earlier this year at the Valley of the Kings in Egypt.
A thorough analysis of the stone can be found in a paper by Vodolazhskaya. The figures in the paper illustrate the impressive precision of the stone’s markings. The sophistication of the geometry employed and the sundial’s ability to measure the passage of time in half hour increments are taken as evidence that ancient Egypt had some influence on the people who inhabited the northern coast of the Black Sea. What strikes me, however, is something that may be unique to an ancient ‘tool’ like this one. In some sense, the intricate geometric structure exhibited by the stone exists only when the stone and the sun are taken together. It emerges from the directed light of the sun together with human perception, thought, and craft. This would be true of any ancient sundial. It just happens that this one made me particularly aware of it, perhaps because of the fine tuning between this one and the latitude of the location where is was found. In the right light, the mathematics isn’t fully isolated from the action of the sun and the sundial maker.
The mathematics isn’t describing the relationship between sunlight, shadow and time, it is the relationship between sunlight, shadow and time. And this is why I thought again about Humberto Maturanna. I think that mathematics, like language, is a kind of shared action, occurring within and among things in the world. It exists in various bodies that may or may not find reason to express it formally.
If you haven’t seen it, the more self-contained mathematical action in the little film Nature by Numbers is worth a look.