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By Joselle, on March 5th, 2012 I asked myself a naive question just the other day: “What is a thought?” I wondered about it when, during a workout, I saw my mind drift, and a chain of unrelated memory fragments were brought to my awareness through spontaneous, even nonsensical associations. Their shared presence was prompted, perhaps, by words or by something else that connected them of which I was unaware. This kind of mental drifting is the justification, in fact, for the free association employed by some psychotherapists to uncover unconscious associations that may be affecting a person’s experience. But, as many of my observations do, it made me think of mathematics. Because, crucial to the development of mathematics is the emergence of relationships among thoughts that first appear to be unrelated.
My question led me, this morning, to look at the new and very ambitious project called the Human Connectome Project. Collaborators on this project explore the complex relationships among billions of neurons that are responsible for the whole of human experience including (a Wall Street Journal article noted) reason, memory and emotion.
I’m not sure why we wouldn’t immediately include imagination.
The word Connectome shares the ome with genome, the entire sequence of an individual’s DNA. The naming anticipates a similarly complete mapping of the hidden biology that somehow makes a particular individual who they are. It is the topic of a recent book and website by MIT scientist Sebastian Seung: Connectome: How the Brain’s Wiring Makes Us Who We Are. Seung spoke at a TED conference in 2010.
The talk only encouraged me to think along the lines I’ve been thinking. Making the sheer vastness of this neuroscientific task clear, Seung explains that there are 100 billion neurons in a human brain, and each one touches many others. He estimates that there are a million times more connections than there are genome letters and, while recalling Pascal’s dread of the infinite, said that the brain is “so awesome in its complexity, it might even be infinite.” So maybe it is also from within that our notions of infinities grow.
These neural connections, or synapses, are responsible for the most fundamental aspects of life itself, movement, respiration and digestion as well as the complexities of human thought. In his talk Seung makes clear that neuron branches that create these connections can both grow and fall away, that synapses themselves can be created or eliminated, grow larger or smaller. The hypothesis is that, with neural activity constantly changing, thinking can change your connectome. It is here, he suggests, that nature meets nurture. He draws an analogy between consciousness and a stream, where neural activity is seen as the water and the connectome as the water’s bed. The stream’s bed, directs the flow of water but is also changed by the water’s flow. We’ve all thought about the stream of consciousness, but perhaps not about how the streams bed can be changed.
There is certainly quite a lot that can be gained from an ideas analysis of mathematics, (like the ones proposed by Rafael Nunez) or the investigations of hardwired quantitative and spatial perceptions (like those of Stanislas Dehaene) But a true appreciation of the nature of mathematics may yet be gained by its resemblance to neural processing itself. Our early observations of quantity and spatial relationships have developed, in a very organic way, into complex mathematical relationships and systems. If one were to try to make a schematic drawing of their advancement it would resemble many things organic, including our own synaptic trees. Numbers, once notches on a bone, become digits that then advance from their correspondence to fingers, to their manifold roles in analyzing complex relationships among sets of abstract objects and abstract spaces, and these eventually lend us images of an increasingly vast cosmos or the probabilities of a subatomic particle’s existence.
It may be that ideas like Max Tegmark’s (that the world is a mathematical object) or the Plantonist position held by many mathematicians (that mathematics is its own independent reality) are inspired by the fact that thought and reality are two sides of the same coin, that they cannot be separated, that the physical processing of thought does not create the thought, but somehow is the thought. It is likely that the pristine abstractness of mathematics makes it one of the most effective mirrors of both the reality we see around us and the reality that is us.
By Joselle, on February 27th, 2012 When I first became interested in studying mathematics an artist friend of mine expressed his disapproval by characterizing mathematicians as people who made bombs. Although I didn’t know very much mathematics at the time, I knew enough to know that he was wrong. But I was reminded today of one of the ways his mistake may have taken shape. George Dyson has written a soon to be released book Turing’s Cathedral: The Origins of the Digital Universe. John Naughton interviews Dyson for The Guardian and characterizes the book in this way:
Turing’s Cathedral is much more than a chronicle of engineering progress: it includes fascinating digressions into the history and physics of nuclear weapons, the fundamentals of mathematical logic, the mathematical insights of Hobbes and Leibniz, the history of weather forecasting, Nils Barricelli’s pioneering work on artificial life and lots of other interesting stuff.
According to Naughton, Dyson
focuses on a small group of mathematicians and engineers working on the hydrogen bomb, led by John von Neumann at the Institute for Advanced Study (IAS) in Princeton, New Jersey (but not at Princeton University), who not only built one of the first computers to realize Turing’s vision of a universal machine, but – more importantly – defined the architectural principles of a general-purpose “stored program computer” on which all succeeding computers were based. Dyson’s argument, crudely summarized, is that the IAS machine should be regarded as the fons et origo of the modern world rather than the ENIAC or Colossus machines that preceded it.
We learn in the interview that running codes developed by von Neumann for the IAS (Institute for Advanced Study) machine were delayed by hardware problems for a couple of years. Under pressure to run bomb calculations, they reconfigured an older machine to run their codes. This, Dyson says. “may have diminished their own prominence as pioneers.” But, he adds, it may be that there was also some reluctance to herald accomplishments associated with the development of the hydrogen bomb.
It seems Dyson also challenges the distinctions made between ‘pure’ and applied mathematics or between the abstract and the pragmatic. From the interview:
JN Another theme that comes over strongly relates to WH Hardy’s famous misconception about the “uselessness” of pure mathematics. You trace very clearly the progression from Hilbert to Gödel to Turing to von Neumann to the IAS machine. My guess is that nobody at the time could have supposed that arguments about the foundations of mathematics would ever have a practical outcome.
GD Yes! It is quite astonishing, for instance, that Turing, who was more or less an outcast, except among a small group of fellow logicians, during the two years he spent in Princeton, was recently voted the second-most influential alumnus of Princeton University (and this from a field going back to 1746!)
But, more consistent with my own concerns are some of the things I read in a piece, by the same title, written by Dyson for Edge in 2005 (which I would expect are now found in the book).
By breaking the distinction between numbers that mean things and numbers that do things, von Neumann unleashed the power of the stored-program computer, and our universe would never be the same. It was no coincidence that the chain reaction of addresses and instructions within the core of the computer resembled a chain reaction within the core of an atomic bomb.
It’s where von Neuman seemed to be going that gets my attention today. Again, from the Dyson piece, a rather long excerpt:
As organisms, we possess two outstanding repositories of information: the information conveyed by our genes, and the information stored in our brains. Both of these are based upon non-von-Neumann architectures, and it is no surprise that von Neumann became fascinated with these examples as he left his chairmanship of the AEC (where he had succeeded Lewis Strauss) and began to lay out the research agenda that cancer prevented him from following up. He considered the second example in his posthumously-published The Computer and the Brain.
“The message-system used in the nervous system… is of an essentially statistical character,” he explained. “In other words, what matters are not the precise positions of definite markers, digits, but the statistical characteristics of their occurrence… a radically different system of notation from the ones we are familiar with in ordinary arithemetics and mathematics… Clearly, other traits of the (statistical) message could also be used: indeed, the frequency referred to is a property of a single train of pulses whereas every one of the relevant nerves consists of a large number of fibers, each of which transmits numerous trains of pulses. It is, therefore, perfectly plausible that certain (statistical) relationships between such trains of pulses should also transmit information…. Whatever language the central nervous system is using, it is characterized by less logical and arithmetical depth than what we are normally used to [and] must structurally be essentially different from those languages to which our common experience refers.”
Or, as his friend Stan Ulam put it,” What makes you so sure that mathematical logic corresponds to the way we think?”
Pulse-frequency coding, whether in a nervous system or a probabilistic search-engine, is based on statistical accounting for what connects where, and how frequently connections are made between given points. As von Neumann explained in 1948: “A new, essentially logical, theory is called for in order to understand high-complication automata and, in particular, the central nervous system. It may be, however, that in this process logic will have to undergo a pseudomorphosis to neurology to a much greater extent than the reverse.”
Von Neumann died just as the revolution in molecular biology, sparked by the elucidation of the structure of DNA in 1953, began to unfold. Life as we know it is based on digitally-coded instructions, translating between sequence and structure (from nucleotides to proteins) exactly as Turing prescribed. Ribosomes and other cellular machinery play the role of processors: reading, duplicating, and interpreting the sequences on the tape. But this uncanny resemblance has distracted us from the completely different method of addressing by which the instructions are carried out.
In a digital computer, the instructions are in the form of COMMAND (ADDRESS) where the address is an exact (either absolute or relative) memory location, a process that translates informally into “DO THIS with what you find HERE and go THERE with the result.” Everything depends not only on precise instructions, but on HERE, THERE, and WHEN being exactly defined. It is almost incomprehensible that programs amounting to millions of lines of code, written by teams of hundreds of people, are able to go out into the computational universe and function as well as they do given that one bit in the wrong place (or the wrong time) can bring the process to a halt.
Biology has taken a completely different approach. There is no von Neumann address matrix, just a molecular soup, and the instructions say simply “DO THIS with the next copy of THAT which comes along.” The results are far more robust. There is no unforgiving central address authority, and no unforgiving central clock. This ability to take general, organized advantage of local, haphazard processes is exactly the ability that (so far) has distinguished information processing in living organisms from information processing by digital computers.
I would like to propose that we shift our focus away from how we can (or cannot yet) imitate what our biology accomplishes, and look, for a moment, at what it is we have found within it. We have traversed territories opened up by our cognizance of number patterns that lead, not only to electronic calculation, but to the creation of electronic images, as well as the simulation of events and experience. What might this tell us about the life of the organism we call the human organism? I will likely follow this up with a look at the von Neumann book.
By Joselle, on February 21st, 2012 I followed a lead today that came at the end of Clifford Pickover’s The Math Book.
The last of Pickover’s 250 milestones in mathematics is Max Tegmark’s Mathematical Universe Hypothesis, which Tegmark published in 2007 in both scientific and popular articles. The hypothesis is that “our universe is not just described by mathematics – it is mathematics. Tegmark is a cosmologist at MIT and scientific director of the Foundational Questions Institute. His NewScientist article from that time, What the Universe if Really Made Of, can be found on his website.
There is also a link on his website to a roundtable discussion that Tegmark joined. The discussion was hosted in 2009 by the Philoctetes Center, a New York City based organization devoted to ‘The Multidisciplinary Study of Imagination.’ It seems the organization lost funding and shut down in 2010. This is unfortunate. I was intrigued by the topic – Mathematics and Religion – and listened for an hour.
The diversity of perspectives was interesting but, notwithstanding the unique focus of the discussion, the group found it difficult to get past the usual questions, about Platonism and the existence of God. And they unexpectedly, spent a fair amount of time arguing about what Spinoza said. This was despite the fact that novelist, philosopher and Harvard Research Associate, Rebecca Newberger Goldstein did a nice job of starting things off. I thought these words of hers opened the door to a number of paths that were never pursued:
Also, in mathematics, although it perceives by proofs, it happens that proofs are often the afterthought. It’s after you’ve already seen it, you have a heuristic grasp of it. Again, if this is a transcendent world, what are we seeing when we’re seeing these things, and how are we making contact with this transcendent world? So – although mathematics itself seems very, very certain, mathematics raises all sorts of questions about human knowledge and the limits of human knowledge, and how we can do what we do when we’re doing mathematics.
Max Tegmark made his argument:
…my guess is that the explanation is that reality is so well described by mathematics because it is ultimately purely mathematical, and not only is our physical world described by a mathematical structure, but it is a mathematical structure. So I think we’re all living in a gigantic mathematical object – not one of the simple ones that we learn about in high school math. We’re not living inside of a cube or a dodecahedron or in the set of integers, but there’s some more complicated mathematical object, maybe M-theory, maybe some – more likely something we haven’t discovered yet which somehow is our reality.
That’s his view in a nutshell – interesting and provocative. What might that imply about mathematics and religion? I don’t think anyone actually took this up. Later in the discussion he made what I thought was a nice observation about time and creation (in mathematics).
Like you asked, for example, a very interesting question of the integers, were they created or have they always existed. That whole question presupposes the existence of time, because you need time for something to first not exist and then exist. A creation event requires time. But we know, and Einstein taught us that there are two ways in which we can think of time. We can think of either a reality being this three-dimensional space where stuff happens over time, or we can think of living in this four-dimensional space-time, a four dimensional space where the fourth dimension has a minus sign in it, which makes it feel like time. And the space-time of Einstein of course, there’s nothing happening in there. If life is a movie then the space-time is like the entire DVD. It’s all in there. The DVD isn’t changing even though there’s all sorts of drama unfolding in the play.
So if you think of our reality as a mathematical object which contains the space-time, then time exists within this mathematical structure, rather than the mathematical structure existing in time.
Earlier in the discussion, Princeton mathematician Edward Nelson made a different observation of time that I liked very much.
Another interesting feature of mathematics is the tremendous time frame. Six is called a perfect number because it’s the sum of its divisors other than itself; one and two and three divide six, one plus two plus three equals six. And Euclid not only did geometry but did number theory in his book and he proved that numbers of a certain form connected with prime numbers were also perfect. And so, yes it was 2000 years later, in the eighteenth century Euler proved that every even perfect number was of Euclid’s form. That’s a tremendous timeframe to be working on the same problem, and it’s a major open problem today, do there exist odd perfect numbers. No one knows.
Nelson writes on whether the completed infinity of all numbers exists in this essay. Goldstein wrote a piece related to the roundtable topic also.
I did enjoy what I finally pulled out of the Spinoza discussion – the idea that God could be imagined as the complete image of what nature is.
But I was surprised to find that there was no opportunity to connect these thoughts to questions about us – how we work, how we’re built, what is the thing we call intuition, what is an image, how we may be motivated, or what it is that we seek. Why is it that mathematics has been seen as pointing to a transcendent reality? There is, after all, a perspective in biology that says that living is cognition – as in Maturana and Varela in The Tree of Knowledge where “cognition is an ongoing bringing forth of a world through the process of living itself.” How are the products of human cognition ‘the process of living itself,’ and what might this mean about religious images? The title of this work in biology certainly suggests that the ideas they present may be related to religious ones.
Despite words like ‘the multidisciplinary study of imagination,’ or Goldstein’s questions about what is it that we’re seeing when we see the idea for which we seek proof, in this discussion of mathematics and religion there was no talk about the organism, only about some of the images that the organism creates. I don’t think it’s possible to have an effective discussion of mathematics and religion without including talk about us.
By Joselle, on February 14th, 2012 I wanted to do something today with math and music, not their formal relationship, but their mingling in our experience. The seed for the thought came from a 2003 MIT paper I found: Music as Embodied Mathematics. The idea explored in this particular paper was inspired by a composition project that involved software designed for college music students.
Indeed, it was in analyzing musically novice students’ accounts of their work-in-progress, particularly as they experimented with rhythmic possibilities, that we noticed mathematical relationships playing a role in their perception and composition of musical coherence.
It may seem unremarkable that the principal mathematics college students spontaneously put to work involved ratio, proportion, fractions and common multiples. However, it turns out that these intuitively generated and perceived music/mathematical relationships are some of the important mathematical concepts that are found to be most problematic for middle school children.
And so they considered that music might provide a way for children to manage these troublesome concepts, and also introduce them to a less tool-oriented view of mathematics.
Engaging both domains together might also enhance the children’s appreciation and understanding of aesthetic relations shared by mathematics and music.
The experiment was small and informal. Researchers met with a group of 6th grade students once or twice a week over a period of three months. But their observations support the idea that the affinity between math and music is multilayered, involving counting and periodicity (repetition), temporal ruler beats (duration) and their proportional relationships, graphic representations (abstraction) and patterned structure.
More subtle and perhaps more interesting intersections between music and mathematics were discovered as the children composed melodies – particularly as the graphic representations helped them to come to consider patterns such as symmetry, balance, grouping structures, orderly transformations, and structural functions. Structural functions include, for instance, pitch/time relations that function to “create boundaries,” or entities (e.g. phrase) some of which sound “incomplete,” thus functioning to resolve or settle on onward motion.
As an aside, I’ve always been intrigued by the sensation of movement in music.
There wasn’t magic in the results of this experiment. Children didn’t just jump into advanced math ideas by thinking about music. But they were thinking about the music and the mathematics, and making sense of them. The authors considered that the idea of a mathematical transformation could be drawn from music, the map of one instance of a given melody, for example, onto another, or one instance of a given drum piece mapped onto another. And they raised the question: “How far can the mathematics of invariance be drawn out of musical experience?” They admit that they have not realized this thought experiment but did report confidently that, in composing their tunes, these 6th graders made use of structural relations that included rule-driven transformations such as sequence, fragmentation and extension by repetition.
I think one of their more important observations is this one:
Perhaps the most general aspect of the affinity between mathematics and music might be the perception and articulate study of patterns. Pursuing this agenda within music might encourage children to become intrigued with looking for patterns in other domains as well. And it might lend a “sense” to mathematics as a tool for understanding more about what we intuitively have some grasp of and care about. (italics my own)
Which brings me to a wonderful little piece by mathematician Marcus du Sautoy that appeared in The Guardian this past June. Du Sautoy quickly gets to the heart of the matter.
Just as notes and rhythms are not all there is to music, so arithmetic and counting are not all there is to mathematics. Mathematics is about structure and pattern. As we’ve explored the universe of numbers, we’ve discovered strange connections and stories about numbers that excite and surprise us. Take the discovery by Fermat, the 17th-century French mathematician, that a prime number that has a remainder of 1 after division by 4 (like 41) can always be written as the sum of two square numbers (41=16+25). It was a realisation that linked the seemingly separate worlds of primes and squares.
Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don’t realise about mathematics is that it involves a lot of choice: not about what is true or false (I can’t make the Riemann hypothesis false if it’s true), but from deciding what piece of mathematics is worth “listening to”.
There is, in everyone’s introduction to math and music, much talk about numbers, counting, and arithmetic. And it is significant that the two disciplines share this. Yet I think one of the most important things that mathematics accomplishes is its linking of “seemingly separate worlds,” perhaps even more that we have yet imagined. Every mathematician and musician knows the vast expanse of surprising structure that lies just beyond the bridge of counting. And this is what Marcus du Sautoy speaks to when he says:
…for me, what really binds our two worlds is that composers and mathematicians are often drawn to the same structures for their compositions. Bach’s Goldberg Variations depend on games of symmetry to create the progression from theme to variation. Messiaen is drawn to prime numbers to create a sense of unease and timelessness in his famous Quartet for the End of Time. Schoenberg’s 12-tone system, which influenced so many of the major composers of the 20th century, including Webern, Berg and Stravinsky, is underpinned by mathematical structure. The organic sense of growth found in the Fibonacci sequence of numbers 1,2,3,5,8,13 . . . has been an appealing framework for many composers, from Bartók to Debussy.
Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry.
Both of the pieces I refer to quote Leibniz when he said:
Music is the arithmetic of the soul, which counts without being aware of it.
By Joselle, on February 6th, 2012 
I read a few articles today that brought aesthetic and religious expression, mathematical curiosity, and physical discovery into contact.
A recent Physics World article reported that an architectural researcher found the first examples of perfect quasicrystal patterns in Islamic architecture. Also known as Penrose tiles, these patterns were described mathematically by Roger Penrose in the mid-1970s.
Quasicrystals are patterns that fill all of a space but do not have the translational symmetry that is characteristic of true crystals. In two dimensions this means that sliding an exact copy of the pattern over itself will never produce an exact match, though rotating the copy will often produce a match. They were first described mathematically by the British academic Roger Penrose in the guise of the famous Penrose tiles. About 10 years later Danny Schechtman of Israel’s Technion University showed that the positions of atoms in a metallic alloy had a quasicrystalline structure. Since then, hundreds of different quasicrystals have been discovered in nature.
In 2007 two physicists in the US reported that they had found an example of a 15th-century geometric pattern in Iran that showed an “almost perfect” example of Penrose tiling. These researchers concluded that the Islamic craftsmen most likely created the patterns using a set of tiles of distinct shapes, each decorated with lines that join to form the final patterns. Several other studies have also suggested that quasiperiodic patterns in Islamic architecture were constructed through local rules such as subdividing or overlapping of tiles. But none of the proposed methods has able to explain how the ancients ended up creating global long-range order in their patterns.
But the new paper identifies three perfect examples of these patterns (one from as early as the 12th century). It also makes an observation that could clarify how they were created.
In this latest work, Rima Ajlouni, an architectural researcher at Texas Tech University in the US, believes that she has identified three examples of quasiperiodic patterns in Islamic architecture without any imperfections.
In her paper, Ajlouni also shows that ancient Muslim designers were able to resolve the complicated long-range principles of quasicrystalline formations. In other words, these designers were fully aware of the extent of connectedness within their work. In all three examples, Ajlouni reconstructs the patterns and shows that the size of a central “seed” figure is proportional to the size of the overall framework of the pattern. She demonstrates that the three patterns could have been created using nothing more than a compass and a straightedge. This construction method that was widespread in Islamic societies to create a variety of media such as woodworks, ceramics and tapestries.
“They were able to create some of the patterns of complex modern mathematics using basic principles alone,” she says.
Some of the inspiration for Penrose’s work came from Kepler’s Monsters. And a New Scientist article on Shechtman’s prize also identifies the quasicrystal’s relationship to another well know mathematical idea:
The quasicrystal is related to the Fibonacci sequence, in which each number is the sum of the two preceding it, such as 1, 1, 2, 3 and so on. The terms in this sequence yield the “golden ratio” tau, and it turns out that tau governs the distances between atoms in quasicrystals.
Another PhysicsWorld article reporting Shechtman’s prize, includes some nice detail.
Finally, an October article in +Plus Magazine explores the relationship between Penrose tiling and Shechtman’s discovery of a quasicrystalline structure.
Penrose was aware of the potential connection to crystallography when he discovered his first non-periodic tilings which had five-fold symmetry in the early 1970s – and that a lattice of atoms following his non-peridoc pattern would disobey the standard rules of crystallography. But creating a Penrose tiling is not straight-forward: most people on their first attempts get stuck with a configuration that can’t be extended any further.
“For this reason I was somewhat doubtful that nature would actually produce such ‘quasi-crystalline’ structures spontaneously,” says Penrose. “I couldn’t see how nature could do it because the assembly requires non-local knowledge. With my tilings you can’t hope to get it right over a large area simply by placing the atoms one after the other where they ‘fit’. At various times a choice would arise and to make the right choice you may have to know what’s going on a long way away at the other end of the structure. If you make the wrong choice you eventually get stuck.”
But it seemed that nature had found a way. The chemist Alan Mackay had used Penrose’s tiles to build a model of a two-dimensional crystal – a layer of atoms sitting at the corners of a Penrose Tiling. This non-periodic crystal would have five-fold symmetry and its diffraction pattern would have the same features as those produced from Shechtman’s crystal. Roger Penrose’s non-periodic tilings of two-dimensions, and their three-dimensional analogues, explained the structure of Shechtman’s impossible crystal.
“Schechtman’s discovery did come as a bit of a surprise to me,” says Penrose. “But as soon as Paul Steinhardt [one of the physicists responsible for making the connection between Mackay’s model and Shechtman’s diffraction pattern] showed me the diffraction patterns that Shechtman had found and that he (Steinhardt) had calculated I was happy to believe that nature had found a way around the problem.”
The mathematics that Penrose investigated gave structure to the raw observation first made by Schechtman. It is the diffraction patterns of an X-ray beam (scattered by the atoms of a substance that was put in its path) that chemists use to construct the three-dimensional structure of a crystalline substance. They look at the symmetries in the diffraction patterns. And the mathematics of group theory provides the language they need.
The 12th century version of Penrose’s tiling idea certainly tells us that there is more than one path to an insight. But the mathematics of this insight, applied to the Schechtman’s observation, is what makes his observation comprehendible. I inevitably think that the role mathematics is playing resembles the nervous system itself, finding structure in data. Daily, the nervous system’s sensory and cognitive processes bring structure to the sensory data that flood the body.
By Joselle, on January 30th, 2012 I have tried to make the argument, in some of the things I have written, that mathematics experiments with the ways we are able to ‘see.’ But there is a great deal of complexity in what it means ‘to see.’ ‘Seeing’ and ‘reasoning’ are not easily unraveled. An infant’s ‘intuitive physics,’ the subject of recent infant cognition research, is also ‘seeing.’
From the point of view of biologist Humberto Maturana and the late Francisco Varela, the nature of the nervous system is understood only within the history of mutually congruent structural changes that happen between systems, or, to make it a little plainer, the interactive history between the cells we identify as a living unit and its containing environment. They call this history of ‘recurrent interactions’ leading to ‘structural congruence,’ structural coupling. This coupling is how they define cognition, even when there isn’t a nervous system. And even when there is one, sensory-motor and cognitive processes cannot be easily disentangled, if at all. In this light, the physicist and the mathematician may be said to experiment as the baby does, spontaneously, but in the presence of a complex history of structural couplings.
One can imagine that the artist experiments with this coupling also, as Alva Noë considers in his piece on Experience and Experiment in Art. Noë looks at the work of artists like Richard Serra. (Photos of some of this work can be found here)
When one first encounters a piece, such as Running Arcs (For John Cage), one is liable to be struck by the scale and the visually inscrutable orientations and distributions of weight. One not only notices these qualities, but one is disturbed by them. One puzzles: what stops these giants from falling over? To wander around or through a piece such as this can cause a loss of balance. In this way the works make us reflect on how we feel, perceptually, in their presence.
And they direct our attention to the complexity of our experience, a complexity we easily overlook. The loss of balance, for example, introduces us to what are strictly non-visual (e.g. vestibular, kinesthetic) components of our ‘visual’ experience.
What I shall argue is that Serra’s work (and also the work of these other artists) enables us to catch ourselves in the act of perceiving and can allow us thus to catch hold of the fact that experience is not a passive interior state, but a mode of active engagement with the world
The weave that the body builds with experience is tight, as is evidenced by cognitive studies like the one showing how body posture can influence our estimations of quantity, reported by Mo Costandi at the Guardian. The study was published here.
According to the mental number line theory, we think of numbers along a horizontal line with small numbers on the left and larger ones to the right. Rather than being specific to certain cultures, this seems to occur universally and may be due to the way in which the brain represents numbers. With this in mind, Anita Eerland and her colleagues hypothesized that secretly making people lean to the left or right would affect their estimations of quantities.
The study shows that the manipulation of body posture influenced the participants’ estimations, even though they were completely unaware of their true posture. How might this work? It is thought that estimating something such as the height of the Eiffel Tower involves a strategy of ‘anchoring and alignment’ – we think of the height of another building and then mentally compare it to the Eiffel Tower. This would involve retrieving information from memory, so leaning to the left may make smaller numbers more accessible than larger ones.
The findings provide further evidence for the embodied cognition hypothesis, the simplest form of which states that the content of the mind is partly determined by the form of the body. We are only just beginning to understand how the body influences mental function, but it now seems quite clear that the influence of the body extends to abstract concepts and – as this new research shows – to complex cognitive processes such as decision-making.
To round out the theme, I find the recent study on how Bowerbirds create illusions in their own sculptures (also reported by Mo Costandi) to be great food for thought.
Male bowerbirds use their intelligence to impress the females, constructing elaborate structures called bowers to attract mates. They are not on master builders, but also accomplished artists. Males of some species decorate their bowers lavishly with flower petals and sparkly manmade objects. The Satin bowerbird even paints the walls of his bower with charcoal or chewed up berries.
Male Great bowerbirds are even more remarkable. Their bowers, which are among the most complex of all, are true marvels of avian architecture. But as well as being builders and artists, males of this species are also magicians – the bowers they build are like a house of illusions, with built-in visual tricks that manipulate females’ perceptions and increase their likelihood of choosing the builder as their mate.
They do so by arranging the objects covering the floor of the court in a particular way, so that they increase in size as the distance from the bower increases. This positive size-distance gradient creates a forced perspective which results in false perceptions of the geometry of the bower, which is visible only to the female when she is standing in the avenue. From her point of view, all of the objects in the court appear to be the same size.
Lest you think it’s not intentional,
Endler and his colleagues manipulated the decorative objects in the bower to reverse the gradient, with the large objects placed closest to the bower and the smaller ones further away, and found that the birds re-arranged the objects to restore the original pattern. This happened very quickly – in all cases, the positive gradient was restored within three days and the pattern was almost identical to the original by two weeks.
There are photos of the structures in Costandi’s blog that are worth seeing. Researchers can’t find reason beyond courtship for these wonderful little experiments with perception. But whatever the reason, these birds are actively engaging their sense of space, and perspective…..making art and mathematics?
By Joselle, on January 23rd, 2012 I wanted to take a closer look at the Lakoff/Nuñez book Where Mathematics Comes From and its relationship to what has come to be called the embodied mind. It seems to me that the biologists who pioneered embodiment had a more radical view of cognition than many of the cognitive scientists who use the paradigm. For example, in one of his papers, Humberto Maturana Romesin, an early explorer of a very broadly defined notion of cognition, has said:
In my view the central theme of cognition is the explanation of experience, not reality, because reality is an explanatory notion invented to explain experience.
This might sound vexingly circular, but this is only because we are so locked into our usual point of view. One of the things that came to mind when I read it, was the difficulty we have with many of the ideas in modern physics – like, for example, the idea that there is no place and time before the big bang, that there is no such thing as nothing, or that virtual particles can pop in and out of existence. These realities are the explanatory notions invented to explain a very broadened experience, the experience that physics provides.
The difficulty with the view presented in Where Mathematics Comes From is that, while it associates mathematics with some very fundamental human experiences, it fails to fully appreciate the way living systems are understood as totalities by many of the biologists whose work inspired embodiment ideas. In particular, it fails to fully appreciate the way living systems exist in relation to their medium and are, in no way, independent of it. As Maturana says:
the behavior that appears is not a feature of the organism, but a condition of its existence in the relational space in which it is a totality, and in which behavior as a relational dynamic involves both the organism and the medium in which it exists.
He goes on to say that one of the reasons we have difficulty with this view:
arises from our cultural training that leads us to think in terms of external causes to explain the occurrence of any phenomenon. This attitude blinds us to the spontaneous nature of all processes in the molecular domain in which we exist. All molecular processes occur spontaneously following a path that arises moment after moment according to the structural dynamics of the different molecules involved.
Where Mathematics Comes From approaches mathematics in a very tool-like way, with an emphasis on how we might be building it. And this is certainly a useful perspective. But it likely misses something because it only minimally addresses the spontaneity of cognitive processes.
In a review of the book back in 2001, James Madden takes note of other deficiencies in the Lakoff/Nuñex perspective:
If I think about the portrayal of mathematics in the book as a whole, I find myself disappointed by the pale picture the authors have drawn. In the book, people formulate ideas and reason mathematically, realize things, extend ideas, infer, understand, symbolize, calculate, and, most frequently of all, conceptualize. These plain vanilla words scarcely exhaust the kinds of things that go on when people do mathematics. They explore, search for patterns, organize data, keep track of information, make and refine conjectures, monitor their own thinking, develop and execute strategies (or modify or abandon them), check their reasoning, write and rewrite proofs, look for and recognize errors, seek alternate descriptions, look for analogies, consult one another, share ideas, encourage one another, change points of view, learn new theories, translate problems from one language into another, become obsessed, bang their heads against walls, despair, and find light. Any one of these activities is itself enormously complex cognitively—and in social, cultural, and historical dimensions as well. In all this, what role do metaphors play?
With a critique more centered on mathematics education, Martin Schiralli and Nathalie Sinclair make a related observation:
WMCF is right about the sensory-motor basis of abstract concepts, but their reduction of abstract concepts to more concrete ones through metaphor fails to explain the fundamental processes involved in acts of abstraction. The very phrases ‘abstract thought’ and ‘abstract concept’ are misleading. The expression that needs to be analysed is ‘thinking abstractly’.
An intuitive framework (more Kantian naturalism than Platonic idealism) might be a given, but later ‘intuitions’ might be the result of the ideational being (the thinker) tapping into the embodied ordering principles and categories that the visceral being (the organism) has been subliminally and experientially processing.
I believe a more careful look at this biological view of mathematics has been explored by Yehuda Rav. Here are some excerpts from his essay on mathematics as seen in the light of evolutionary epistemology.
Thus, Maturna (1980, p. 13) writes: “Living systems are cognitive systems, and living as a process is a process of cognition”. What I wish to stress here is that there is a continuum of cognitive mechanisms, from molecular cognition to cognitive acts of organisms, and that some of these fittings have become genetically fixed and are transmitted from generation to generation. Cognition is not a passive act on the part of an organism, but a dynamic process realized in and through action.
When we form a representation for possible action, the nervous system apparently treats this representation as if it were a sensory input, hence processes it by the same logico-operational schemes as when dealing with an environmental situation. From a different perspective, Maturana and Varela (1980, p. 131) express it this way: “all states of the nervous system are internal states, and the nervous system cannot make a distinction in its process of transformations between its internally and externally generated changes.”
Thus, the logical schemes in hypothetical representations are the same as the logical schemes in coordination of actions, schemes which have been tested through eons of evolution and which by now are genetically fixed.
As it is a fundamental property of the nervous system to function through recursive loops, any hypothetical representation which we form is dealt with by the same ‘logic’ of coordination as in dealing with real life situations. Starting from the elementary logico-mathematical schemes, a hierarchy is established. Under the impetus of socio-cultural factors, new mathematical concepts are progressively introduced, and each new layer fuses with the previous layers. In structuring new layers, the same cognitive mechanisms operate with respect to the previous layers as they operate with respect to an environmental input. …..The sense of reality which one experiences in dealing with mathematical concepts stems in part from the fact that in all our hypothetical reasonings, the object of our reasoning is treated by the nervous system by means of cognitive mechanisms which have evolved through interactions with external reality.
Mathematics is a singularly rich cognition pool of mankind from which schemes can be drawn for formulating theories which deal with phenomena which lie outside the range of daily experience, and hence for which ordinary language is inadequate.
I would like to add one further note. Near the beginning of his essay, Rav says this:
mathematics and objective reality are related, but the relationship is extremely complex and no magic formula can replace patient epistemological analysis.
And later:
The nervous system is foremost a steering device for internal and external coordination of activities.
For me the most provocative thing about mathematics is what it may be telling us about the connectedness of the internal and the external – experiences that we often have great difficulty reconciling.
By Joselle, on January 9th, 2012 I listened to three short talks today and found that they had something nice in common – they each show us how sensory experience (often vision) gives rise to mathematics that provides access to what cannot be seen, and clarifies what is seen.
The first of these talks was called Symmetry, reality’s riddle presented by Marcus du Sautoy.
It begins with a description of Galois’ famous death in a duel. Galois stayed up the whole night before the duel writing letters and outlining his mathematical ideas. Marcus du Sautoy explains how Galois had found a new language, the language of symmetry, and goes on to take note of the ubiquitous presence of symmetry, showing it to us in, among other things, molecular structure, particle physics and art. Du Sautoy explains how Galois saw that it is not just the symmetries we see in an object that characterize the object, but also the way the symmetries of an object interact with each other. With this insight, he develops a language that can identify the substance of the things unseen. The symmetry that underlies a physical object is captured by a number. And du Sautoy makes the point that this language, expressed in number, now makes it possible for him to create symmetrical objects in high dimensional spaces. This is the mathematical idea of group theory.
The second was a talk given by David Deutsch: A new way to explain explanation. While I didn’t exactly enjoy his critique of pre-scientific ideas, he made some clarifying points about empiricism and the scientific method. He observes that science is originally distinguished by the conviction that all knowledge is derived from the senses. While, he says, this helped by promoting observation and experiment, “it was obvious that there was something horribly wrong with it.”
Empiricism is inadequate because, well, scientific theories explain the seen in terms of the unseen. And the unseen, you have to admit, doesn’t come to us through the senses. We don’t see the origin of species. We don’t see the curvature of space-time, and other universes. But we know about those things. How?
What we see, in all these cases, bears no resemblance to the reality that we conclude is responsible – only a long chain of theoretical reasoning and interpretation connects them
Deutsch makes the argument that testability is only valuable when the theory to be tested is “hard to vary, because every detail plays a functional role.” By way of an example, he identifies all of the functional relationships in the modern explanation of the seasons – “that surfaces tilted away from radiant heat are heated less, that a spinning sphere, in space, points in a constant direction, that tilt also explains the sun’s angle of elevation at different times of year, and that it predicts that the seasons will be out of phase in the two hemispheres.”
Finally, it was Benoit Mandelbrot’s talk: Fractals and the art of roughness that I most enjoyed.
Mandelbrot also found a number that characterized an observation, a number that denoted the roughness of a surface. In an engaging informal style he just tells us stories.
Humanity had to learn about measuring roughness…..Very few things are very smooth…What’s the length of the coastline, which seems to be so natural because it’s given in many cases, is, in fact a complete fallacy; there’s no such thing. You must do it differently.
What good it that, to know these things? Well, surprisingly enough, it’s good in many ways…Now a lung is something very strange…The volume of a lung is very small, but what about the area of a lung?…Anatomists were arguing very much about that. Some say that a normal male’s lung has an area of the inside of a basketball court. And others say, no, five basketball courts…The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical consideration: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, or a man and or a little rodent….And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all of these branching systems, for which there is no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry.
Mandelbrot very gently makes clear that the objects conjured up by 19th century mathematicians, functions that have been called monstrous and used to demonstrate the break between mathematics and visible reality, are now used to describe some aspects of nature’s complexity. He shows us the appearance of a “fractal-to-be” in an 18th century Japanese painting and notes the intuitive grasp of fractals in the work of engineer Gustave Eiffel and in the tower that bears his name. He tells us that halfway through his career he decided to test himself. “Could I just look at something which everybody had been looking at for a long time and find something dramatically new?” So he looked at “these things called Brownian motion,”
Then I was telling my assistant, “I don’t see anything. Can you paint it?” So he painted it, which means he put inside everything. “Well, this thing came out…,” he said. And I said, “Stop! Stop! Stop! I see; it’s an island.”
Mandelbrot could see that this island had a fractal dimension but it was his friends who, 20 years later, won the Fields medal when they proved it.
Here, again, a sensory idea (roughness) is characterized by a number which takes some things away (like the length of a coastline) and brings other things into view (like a refined understanding of the lungs).
This talk is very pleasant to listen to and I recommend it if you haven’t seen it. Mandelbrot gives a sketchy description of the Mandebrot set but successfully describes the simplicity of its starting point. And he concludes with these very nice words:
Bottomless wonders spring from simple rules, which are repeated without end.
By Joselle, on January 3rd, 2012 Unfortunately for us, philosophies of science and mathematics are rarely brought to the attention of individuals who are not engaged in these efforts. Yet, while difficult to access, the views of the world provided by mathematics and science are pregnant with meaningful implications for all of us. I have always been struck by the depth of reflection in Hermann Weyl’s writing. These are thoughts that come clearly from a person, an individual working to reconcile all of the images of himself and the world that have taken shape in his mind’s eye.
At the end of his essay The Unity of Knowledge he summarizes that at the basis of knowledge is (1) intuition “the mind’s originary act of seeing what is given to him,” (2) expression, “the active counterpart of passive understanding” and (3) Thinking the possible, or imagination (he then apologizes for some of his indecision of mind) For me, these three things are mathematics. And seeing mathematics in this way tells us something about ourselves and the extent of our potential.
I’ve pulled out other excerpts that I found particularly provocative from a collection of selected Weyl pieces, edited by Peter Pesic, entitled Mind and Nature.
The first is from The Open World published in 1932. It is a reflection on the non-causal aspect of quantum theory, which Weyl sees as bridging organic and inorganic nature.
According to vitalism the living organism reacts as a whole; its functions are not additive. The manner in which its structure is preserved throughout growth, in spite of a outside influences and perturbations, is not to be explained by small scale causal reactions between the elementary parts of the organism. Now we see that according to quantum physics the same applied even to inorganic nature and is not peculiar to organic processes. It is out of the question to derive the state of the whole from the state of its parts. This leads to conditions which may most plainly if not most correctly be interpreted as a peculiar non-causal “understanding” between the elementary particles, that is prior to and independent of the control exercised by differential laws which regulate probabilities…It seems therefore that the quantum theory is called upon to bridge the gap between inorganic and organic nature; to join them in the sense of placing the origin of those phenomena which confront us in the fully developed organism as Life, Soul and Will back in the same original order of nature to which atoms and electrons also are subject. So today less than ever do we need to doubt the objective unity of the whole of nature…..
Within a lengthy discussion of the concept of infinity in mathematics he makes the following remarks:
Mathematics is not the rigid and uninspiring schematism which the layman is so apt to see in it; on the contrary, we stand in mathematics precisely at the point of intersection of limitation and freedom which is the essence of man himself.
Something entirely new takes place when I embed the actually occurring number symbols in the sequence of all possible numbers…The given is embedded in the ordered manifold of the possible, not on the basis of descriptive characteristics, but on the basis of certain mental or physical operations and reactions to be performed on it – as, for example, counting.
Consider the definition “n is an even or an odd number according as there exists or does not exist a number x for which n = 2x. For one who accepts this with its appeal to the infinite totality of numbers x as having a meaning, the sequence of numbers open into infinity has transformed itself into a closed aggregate of objects existing in themselves, a realm of absolute existence which “is not of this world,” and of which the eye of our consciousness perceives but reflected gleams.
We reject the categorical finiteness of man…..On the contrary, mind is freedom within the limitations of existence; it is open toward the infinite…The completed infinite we can only represent in symbols. From this relationship every creative act of man receives its deep consecration and dignity. But only in mathematics and physics, as far as I can see, has symbolical-theoretical construction acquired sufficient solidity to be convincing for everyone whose mind is open to these sciences.
The last one is from the piece called Mind and Nature (1934).
The impossibility of designing a picture of reality other than on the background of possibility appears to be founded on the circumstance that existence is a penetration of the what and the how, and consequently arises from a contact of object and subject, of pure factuality and freedom.
I was particularly taken by the words, “existence is a penetration of the what and the how, and consequently arises from a contact of object and subject” Penetration is not a word one might expect to use to characterize our existence, but I think it leads to the right images – like probe, permeate, immerse, sink into, as well as fathom and understand.
In his introduction Pesic makes the point that Weyl’s praise of the symbol “includes the mathematical no less than the literary, artistic, and poetic.” He tells us that
Weyl gained perspective, insight, and altitude by thinking back along the ever-unfolding past and studying its great thinkers, whom he used to help him soar, like a bird feeling the air under its wings.
By Joselle, on December 27th, 2011 Hearing about visual processes, from neuroscientists and artists alike, consistently brings mathematical thoughts to mind for me – like Samir Zeki’s descriptions of how visual images are constructed, or the Impressionist painters’ attention to the sensations in the eye rather than the subject of the painting, and, of course, Poincaré’s suggestion that visual space has more than three dimensions.
The relationship between vision and mathematics (or sensation and symbolic thought) came up again today, when I followed the links in a December 15 Vision Help Blog. The blog begins with a reference to an Oliver Sacks story about a blind individual given sight, and concludes with a paper pointing to the psychological significance of the reluctance of organ donors to donate corneas.
The Sacks vignette tells us a lot about vision, and explains why freeing the cornea of obstruction doesn’t just give us sight. We have to learn to see. We have to learn how the many appearances of an object belong to the one object.
We achieve perceptual constancy — the correlation of all the different appearances, the transforms of objects — very early, in the first months of life. It constitutes a huge learning task, but is achieved so smoothly, so unconsciously, that its enormous complexity is scarcely realized (though it is an achievement that even the largest supercomputers cannot begin to match).
It is for this reason that the subject of Sacks’ story who, at age 50, was given back the sight he had lost as a very young child, could not integrate the various pieces of visual stimuli into a coherent image.
…with half a century of forgetting whatever visual engrams he had constructed, the learning, or relearning, of these transforms required hours of conscious and systematic exploration each day. This first month, then, saw a systematic exploration, by sight and touch, of all the smaller things in the house: fruit, vegetables, bottles, cans, cutlery, flowers, the knickknacks on the mantelpiece — turning them round and round, holding them close to him, then at arm’s length, trying to synthesize their varying appearances into a sense of unitary objecthood. […]” (italics added)
While I cannot make any precise correspondence, I do see the visual processing mechanisms that this individual had not developed as ‘the mathematics’ of sight, as the way to capture the many facets of any sensation into a recognizable whole, one that the mind can manage. Even some of the words used in the story call to mind things like algebra and geometry:
An infant merely learns. This is a huge, never-ending task, but it is not one charged with irresoluble conflict. A newly sighted adult, by contrast, has to make a radical switch from a sequential to a visual-spatial mode, and such a switch flies in the face of the experience of an entire lifetime. (italics added)
But there is also a detail in the story that points to what I see as the body’s sheer tenacity, the tenacity to be, however that is accomplished. Everyone was surprised when one of the first things Virgil could recognize was a symbolic thing – it was the alphabet:
Virgil’s first formal recognitions when the bandages were taken off had been of letters on the ophthalmologist’s eye chart…How was it that he had so much difficulty recognizing faces, or the cat, and so much difficulty with shapes generally, and with size and distance, and yet so little difficulty, relatively, recognizing letters? When I asked Virgil about this, he told me that he had learned the alphabet by touch at school, where they had used letter blocks, or cutout letters, for teaching the blind. I was struck by this and reminded of [R.L.] Gregory’s patient S.B.: “much to our surprise, he could even tell the time by means of a large clock on the wall. We were so surprised at this that we did not at first believe that he could have been in any sense blind before the operation.” But in his blind days S.B. had used a large hunter watch with no glass, telling the time by touching the hands, and he had apparently made an instant “cross-modal” transfer, to use Gregory’s term, from touch to vision. Virgil too, it seemed, must have been making just such a transfer. […]
I remembered reading about a blind mathematician, and today found more about him in a 2002 AMS article called The World of Blind Mathematicians. In this story, representations created by touch are also noted:
A visitor to the Paris apartment of the blind geometer Bernard Morin finds much to see. On the wall in the hallway is a poster showing a computergenerated picture, created by Morin’s student François Apéry….The surface plays a role in Morin’s most famous work, his visualization of how to turn a sphere inside out. Although he cannot see the poster, Morin is happy to point out details in the picture that the visitor must not miss.
Later Morin brings out clay models that he made in the 1960s and 1970s to represent shapes that occur in intermediate stages of his sphere eversion.
The models were used to help a sighted colleague draw pictures on the blackboard…It is startling to consider that such a precise, symmetrical model was made by touch alone. The purpose is to communicate to the sighted what Bernard Morin sees so clearly in his mind’s eye.
And this:
Morin recalled that, when a sighted colleague proofread Morin’s thesis, the colleague had to do a long calculation involving determinants to check on a sign. The colleague asked Morin how he had computed the sign. Morin said he replied: “I don’t know—by feeling the weight of the thing, by pondering it.”
Morin believes there are two kinds of mathematical imagination. One kind, which he calls “time-like”, deals with information by proceeding through a series of steps. This is the kind of imagination that allows one to carry out long computations. “I was never good at computing,” Morin remarked, and his blindness deepened this deficit. What he excels at is the other kind of imagination, which he calls “space-like” and which allows one to comprehend information all at once.
Space-like imagination is tactile as well as visual:
One thing that is difficult about visualizing geometric objects is that one tends to see only the outside of the objects, not the inside, which might be very complicated. By thinking carefully about two things at once, Morin has developed the ability to pass from outside to inside, or from one “room” to another. This kind of spatial imagination seems to be less dependent on visual experiences than on tactile ones. “Our spatial imagination is framed by manipulating objects,” Morin said. “You act on objects with your hands, not with your eyes. So being outside or inside is something that is really connected with your actions on objects.” Because he is so accustomed to tactile information, Morin can, after manipulating a hand-held model for a couple of hours, retain the memory of its shape for years afterward.
Another anecdote in the article refers to mathematician Norberto Salinas
In a contribution to a Historia-Mathematica online discussion group about blind mathematicians, Eduardo Ortiz of Imperial College, London, recalled examining Salinas in an analysis course at University of Buenos Aires. Salinas communicated graphical information by drawing pictures on the palm of Ortiz’s hand, a technique that Ortiz himself later used when teaching blind students…
I particularly like this suggestion:
…blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.
Happy New Year!
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