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By Joselle, on October 17th, 2011 I came across an article on vision at Physicsworld.com that is concerned chiefly with how digital imaging technology may and may not be able to provide a fix for damaged retinas. While digital cameras can better mimic the human eye, the gaze of the camera, unlike the eye, is static and uniform.
As sensors in digital cameras fast approach the 127 megapixels of the human eye, clinical trials are under way to implant this technology directly into the retina. But Richard Taylor cautions that such devices must be adapted for humans, because of the special nature by which we see.
Historically,
even the best optical theories suffered from a weakness: given that light rays bounce off a friend’s face, why can we not spot it immediately in a crowd – even though it is directly before our eyes? We are forced to conclude that the visual system is not passive but that it has to hunt for the information we need.
The eye does not employ the Euclidean design of cameras. In other words, the eye’s photoreceptors are not arranged in a uniform, 2-dimensional array across the retina. Instead the eye’s seven million cones are “piled into the central region of the retina.” The eye has to move if the focus of our attention is to fall mainly on the fovea, which, while being only 1% of the retina, uses more than 50% of the visual cortex. Richard Taylor and his colleagues at the University of Oregon investigated this movement or how we search for information in a complex scene. By tracking the motion of the eye they found that
the eye searches one area with short steps before jumping a larger distance to another area, which it again searches with small steps, and so on, gradually covering a large area.
It turned out that the trajectory of the gaze is like a fractal, a 1-dimensional line that starts to occupy a 2-dimensional space because of its repeating structure. Weierstrass’ example of a function which is everywhere continuous but nowhere differentiable is actually a fractal.
One of the intriguing properties of a fractal pattern is that its repeating structure causes it to occupy more space than a smooth 1D line, but not to the extent of completely filling the 2D plane. As a consequence, a fractal’s dimension, D, has a value lying between 1 and 2. By increasing the amount of fine structure in the fractal, it fills more of a 2D plane and its D value moves closer towards 2.
The article explains how the D value is calculated and found that eye movements traced out fractal patterns with a D value of 1.5, which mimics the foraging patterns of an animal’s search for food.
Significantly, fractal motion (figure 1d, middle) has “enhanced diffusion” compared with Brownian motion (figure 1d, right), where the path mapped out is, instead, a series of short steps in random directions. This might explain why a fractal trajectory is adopted for both an animal’s searches for food and the eye’s search for visual information. The amount of space covered by fractal trajectories is larger than for random trajectories.
In this light, the author raises an interesting question:
what happens when the eye views a fractal pattern of D = 1.5? Will this trigger a “resonance” when the eye sees a fractal pattern that matches its own inherent characteristics?
Experimental collaborations between psychologists and neuroscientists found that images matching the fractal dimension of the eye’s searching movement are ones that are most aesthetically pleasing. And, even more provocative, that exposure to these images can reduce our physiological responses to stress by 60%. Looking further,
preliminary functional-magnetic-resonance-imaging (fMRI) experiments indicate that mid-D fractals preferentially activate distinct regions of the brain. This includes the parahippocampal area, which is associated with the regulation of emotions such as happiness.
Examples of these 1.5-dimensional fractals are images of clouds, trees and river patterns (likely golf course landscaping). This contrasts with the perspective of some evolutionary psychologists that the surprising cross-cultural appreciation of the beauty of particular landscapes is grounded in an ancestral ideal of the Pleistocene savannas.
But, more importantly, it is an indication of the way our bodies and our thoughts are like two sided mirrors, or perhaps how the internal and the external inevitably match. And, for me, it says again how mathematics can search out the forms they share.
By Joselle, on October 10th, 2011 A recent Radiolab episode brought some interesting things together by exploring loops, repetitions, and self-referencing phenomena.
Among other things, they told the story of Melanie Thernstrom (The Pain Chronicles) who, in trying to manage her pain, investigated the self-inflicted pain of religious rites. She later did some work with neuroscientist Sean Mackey. Mackey had seen how pain can be affected by changes in ones state of mind and he considered that if an individual could somehow watch the pain signaling activity, perhaps they could control it. “All of our pain is in our head,” he says definitively. Pain is a conversation between the brain and the body.
Mackey set Melanie up in an MRI and gave her access to a view of her brain. The part of the brain she watched is known to be involved with pain perception, and with “turning pain up and down.” To make what she was seeing comprehensible, the brain activity was illustrated with a visual metaphor, a fire. Melanie describes her effort, her failed attempts, and how she was finally able to do it. But she can’t quite do it without the visual feedback that she was getting from her work with Mackey. It seems that something significant is gained by being able to look at what the brain (or the body) is doing, and for this to be something we can understand.
It was the visual component of this loop that I found most intriguing – the fact that one of the ways we can gain access to an unconsious internal mechanism is by representing it. This is the way I think about mathematics – as the represention of internal mechanisms that allows our conscious participation in how the body can know, anything.
Steve Strogatz adds a specifically mathematical experience to this episode with some talk about Gödel’s Incompleteness Theorem. He described how mathematicians and philosophers struggled with what has come to be called the barber paradox, or the statement that in a particular town, the barber shaves everyone who doesn’t shave himself. Can the barber shave himself? Who shaves him? Russell teased this kind of problem out of set theory (as if it was an unfortunate tangle) but later Gödel was lured more deeply into these paradoxical loops. He probed mind boggling claims like: “This statement is a lie.” If it’s true then it’s false and if it’s false then it’s true. A careful analysis of the problem led him to his famous Incompleteness Theorem, definitively establishing that certain things in mathematics are simply undecidable. And so there may be something that is true but cannot actually be proven.
It was suggested that Gödel found this liberating because, as Strogatz states it, “There was profound mystery forever.” All wasn’t mechanical.
Janna Levin,who recently wrote a book about Gödel, also spoke on the program. I found an interview with her online where she says some interesting things. Like this for example:
We mine our own mind, and that’s how we uncover these seemingly external truths. Gödel only believed in the mind part. He wasn’t so sure about the external reality part, and he often said things like — I think I even saw it on the Simply Gödel website – that “I don’t believe in natural science.” And I think what he meant by that is he just wasn’t sure about external reality. He was really doubtful of it, which is so strange. But he really believed that mathematical concepts existed, they were real, and that the mind migrated to this pure, platonic reality, and migrated over reincarnations. He really believed in transmigration of the soul. So he was pretty out there.
When she talks about the choices she made for how to write her story of Gödel and his insight she says:
I could do more to create an impact of that idea, on the sense of the feeling that hits you in your solar plexus, of what it means to know something’s true, but can’t prove it to be true. And to only be able to approach truth, and yet for it to be your obsession.
We mine our own mind, and that’s how we uncover these seemingly external truths. Gödel only believed in the mind part. He wasn’t so sure about the external reality part……………It’s probably all the same thing…
By Joselle, on October 3rd, 2011 I feel like I was pulled into a little whirlpool of interesting bits of info this morning. I was attracted to the title of David Castelvecchi’s blog: Archimedes and Euclid? Like String Theory versus Freshman Calculus. The blog reports the opening of an exhibition at the Walters Art Museum in Baltimore, showcasing one of three medieval copies of Archimedes’ works, the Archimedes Palimpsest. The first interesting bit is that the text, which had been erased and recycled into a prayer book, was brought back to life with the use of a particle accelerator!
As the exhibition will display on panels and videos, imaging experts were able to map much of the hidden text using high-tech tools—including x-rays from a particle accelerator—and to make it available to scholars.
As Castelvecchi describes this treasure, he takes note of some of its significance as interpreted by Reviel Netz at Stanford University:
Reviel Netz, a historian of mathematics at Stanford University, discovered by reading the “Method of Mechanical Theorems” that Archimedes treated infinity as a number, which constituted something of a philosophical leap. Netz was also the first scholar to do a thorough study of the diagrams, which he says are likely to be faithful reproductions of the author’s original drawings and give crucial insights into his thinking.
An LA Times article (linked to the Palimpsest website itself) contains a similar observation from Netz:
The X-ray image also revealed a section of “The Method” that had been hidden from Heiberg in the fold between pages. It contained part of a discussion on how to calculate the area inside a parabola using a new way of thinking about infinity, Netz said. It appeared to be an early attempt at calculus — nearly 2,000 years before Isaac Newton and Gottfried Wilhelm Leibniz invented the field.
One passage he studied several years ago involved the innumerable slices and lines that could be made from a triangular prism similar to a wedge of cheese. Netz said the passage, which was unreadable to Heiberg, showed that Archimedes was grappling with the concept of infinity long before other mathematicians.
Archimedes’ use of infinite processes, in addressing the value of pi for example, is well known. But there are two details that get my attention in Netz’s evaluation of these ancient works. One is that Archimedes imagined the numerical nature of infinity and the other is the perhaps newly appreciated visual aspect to his work.
Reviel Netz is a professor of classics and specialist in ancient mathematics and cognitive history at Stanford University. In the book he wrote, The Archmedes Codex, Netz makes the argument that ancient Greek mathematics was visual in a way the modern reader of science and mathematics may fail to appreciate. For the modern reader, mathematics ideas may be illustrated visually, but they are not established visually. But Netz explains that the drawings of Archimedes are not illustrations, they are arguments.
Ancient diagrams are schematic, and in this way they represent the broader, topological features of a geometric object. Those features are indeed general and reliable; a diagram represents them just as well as language represents them. And so, ancient diagrams can form part of the logic of an argument which is perfectly valid.
We have learned, therefore, something crucial and surprising about Archimedes’ thought process, about his interfaces. He essentially relied on the visual; he used schematic diagrams that can be used in perfect logical rigor without danger of error based on visual evidence. When Archimedes gazed at his diagrams along the Syracusan seashore…..I know that what he saw there was a crucial part of his thought process—one of the most basic tools that made Greek science so successful.
The entire discussion of Archimedes, inspired by this unexpected find, brings mathematics back, again, to the intuitive, even the sensory. Archimedes was an extraordinary thinker, but his ancient world magnifies, for me, the very internal nature of his investigation.
There is a NOVA piece on the discovery of the hidden manuscript and on that page a link to a piece on infinity.
This piece contains a number of paintings by Quita Brodhead and contains the following note:
No fear of the infinite: Quita Brodhead was an abstract painter who investigated infinity in her “Endless Circle” series and other paintings, several of which illustrate this article. Brodhead died in September 2002 at the age of 101.
You can listen to Quita Brodhead herself here. I recommend it.
By Joselle, on September 26th, 2011 I recently read the announcement of a National Science Foundation Career Award given to Mariel Vazquez (Associate Professor at San Francisco State University) for the work she does in mathematics and biology. Vazquez has been involved for some time in the application of knot theory to the analysis of DNA.
Knot Theory is one of those things that associates a mathematical idea with an early cognitive development, in this case our recognition of the value of the spatial manipulation a string of some sort, and our ability to make one. It’s a skill associated with a turning point in our evolution. We found a way to secure objects, make jewelry, or display something, (like the numerical information in the quipu knots of the Inca Empire).
We also found meaning in these spatial twists, that we may no longer fully understand, manifest in the knotted images of Celtic monks in the Book of Kells. A nice history of what we now call knot theory can be found here.
The 19th century consideration, in physics, that atoms were knotted tubes of ether, actually inspired questions about equivalences among knots, or questions about how to determine when one of them could be made to look like the other. But after it was clearly established that there was no ether, the knot theory of atoms became obsolete. Yet mathematicians continued to be interested in the analysis of sameness among knots, and a significant turning point in this investigation happened when it was discovered that knot equivalence was best understood by looking at the complement of a knot, the space around it that is not the knot itself. Knots’ relationship to other topological considerations was then better understood.
Topology is a kind of geometric thinking, where traditional shapes (like circles, squares, triangles) are indistinguishable. The topological properties of a bowl are equivalent to a disc, and a disc can be reduced to a point. Objects have no rigidity in topology. They can be stretched and shrunk. Distance considerations are irrelevant. Perhaps the flexibility of this kind of thinking gives the mind the chance to focus on other things. And knot theory’s association with this conceptual framework has given it ever-broadening application, as is evidenced by Vazquez’s work in biology.
Mariel Vazquez became fascinated with mathematics and biology early, in high school. She was drawn to work in pure mathematics, but didn’t see the way she could bring this interest to biology. Later, however, as an undergrad, she attended talks on DNA topology and found what she needed. In the article, knot theory’s relevance to DNA is explained in this way:
“When DNA is packed into a cell it doesn’t look like the straight double helix that we see in textbooks pictures,” Vazquez said. “In order to fit into the cell, the double helix is twisted and coiled around itself and around proteins.”
One of nature’s problems is that the two strands of the DNA’s double helix must be separated and unwound in order to be copied, allowing the genome to replicate. Scientists have found enzymes which disentangle DNA, allowing it to replicate, but much is still to be learned about how they work.
In particular:
Vazquez and her colleagues at Oxford University are studying the action of an enzyme that disentangles DNA in the bacterium Escherichia coli (E. coli). “We’ll use mathematics to study the enzymatic mechanism and computer simulations to determine the most probable pathways of disentanglement,” Vazquez said.
It’s also nice to see this young mathematician’s interest in reaching out. She is planning activities that introduce DNA topology to the general public and has already begun introducing knot theory notions to children as young as six years old.
“I can do it, I know that it works,” said Vazquez, who has run a math club for first- and second-grade children over the last year. “I used ropes, ribbons, glue, bubbles and computers to teach them about knot theory, and the children loved it,” she said.
You can hear her speak about her work in a video interview which can also be accessed at the end of the SF State article.
And I found a website designed to give a very accessible description of knot theory and its relevance to DNA studies here.
By Joselle, on September 19th, 2011 In a recent post on the Scientific American blog network, George Musser reported on talks given by neuroscientists at a conference, organized by the Foundational Questions Institute on how the brain works to construct our sense of past, present and future.
Musser’s post made some observations that were familiar to me – like the idea that when we retrieve a memory, we also rewrite it since memories are made from “a smattering of impressions we weave together into what feels like a seamless narrative.” And so we’re actually always retrieving the last one we recollected, calling the accuracy of any memory into question.
But Musser also made mention of work I knew nothing about that uses electric fish to understand something about sensation. Malcolm MacIver has taken note of the fact that when one of these fish species is hunting, it swims at a thirty-degree angle to its body which, while tripling the water drag, also increases the volume of water sensed by its electric field. For these fish the volume of the region it can scan is about equal to the volume of the region it can reach directly. They’ve found a way to maximize the range of their perceptions. But MacIver goes on to explain that for land animals, given how far light can travel, what we can see extends far beyond our immediate grasp. MacIver imagines that this is directly related to the development of consciousness since it gave our ancestors the opportunity to plan and deliberate before making a move. I have also thought that changes in primate vision are related to the development of the analytic and deliberative skills that are our nature but I’ve never used electric fish to think about it.
In the end, the observation I found most interesting was the one Musser makes about the talk by David Eagleman. Eagleman demonstrated that our consciousness lags 80 milliseconds behind actual events. Musser explains:
the brain tries to reconstruct events retroactively and occasionally gets it wrong. The reason, he suggested, is that our brains seek to create a cohesive picture of the world from stimuli that arrive at a range of times. If you touch your toe and nose at the same time, you feel them at the same time, even though the signal from your nose reaches your brain first. You hear and see a hand clap at the same time, even though auditory processing is faster than visual processing. Our brains also paper over gaps in information, such as eyeblinks.
There are some interesting details cited:
The 80-millisecond rule plays all sorts of perceptual tricks on us. As long as a hand-clapper is less than 30 meters away, you hear and see the clap happen together. But beyond this distance, the sound arrives more than 80 milliseconds later than the light, and the brain no longer matches sight and sound. What is weird is that the transition is abrupt: by taking a single step away from you, the hand-clapper goes from in sync to out of sync. Similarly, as long as a TV or film soundtrack is synchronized within 80 milliseconds, you won’t notice any lag, but if the delay gets any longer, the two abruptly and maddeningly become disjointed. Events that take place faster than 80 milliseconds fly under the radar of consciousness. A batter swings at a ball before being aware that the pitcher has even throw it.
The cohesiveness of consciousness is essential to our judgments about cause and effect—and, therefore, to our sense of self. In one particularly sneaky experiment, Eagleman and his team asked volunteers to press a button to make a light blink—with a slight delay. After 10 or so presses, people cottoned onto the delay and began to see the blink happen as soon as they pressed the button. Then the experimenters reduced the delay, and people reported that the blink happened before they pressed the button.
But, more importantly, Eagleman also points out that physics is built “on top of our intuitions,” and that it begins with sense data. Musser elaborates by saying that physics is built “on a recognition of the limits of perception.”
The whole point of theories such as relativity is to separate objective features of the world from artifacts of our perspective. One of the most important books of the past two decades on the physics and philosophy of time, Huw Price’s Time’s Arrow and Archimedes’ Point, argues that concepts of cause and effect derive from our experience as agents in the world and may not be a fundamental feature of reality.
What I find striking about the whole discussion is that, except for this reference to Archimedes, no mention is made of the role mathematics plays in this endeavor to “separate objective features of the world from artifacts of our perspective.” Certainly it is true that mathematics is not physics. But their tight relationship should, at the very least, provide the opportunity to consider the way the mathematics is used to disarm our mental habits, prejudices, and expectations, or how it is that mathematics can reveal sameness, equivalence, where none is perceived. One of the best pieces of information for me from Musser’s post is the presence of the Foundational Questions Institute Community where an initiative is driven by individuals asking a number of questions in physics and cosmology. Among their questions are these:
What is the relationship between physics, mathematics and information? What determines what exists? How “real” is the world of mathematics—and how “real” is the world of matter?
Why does the universe seem so complex, given its simple initial conditions, and the elegant mathematics that describes it? Is life ubiquitous in the universe (or beyond)? How does matter give rise to consciousness—or does it?
The argument is often made that, for the average individual, the value of mathematics can best be seen when we use it to calculate our savings or our mortgages. A case is point is this suggestion in a recent NYTimes article on math education:
In math, what we need is “quantitative literacy,” the ability to make quantitative connections whenever life requires (as when we are confronted with conflicting medical test results but need to decide whether to undergo a further procedure) and “mathematical modeling,” the ability to move practically between everyday problems and mathematical formulations (as when we decide whether it is better to buy or lease a new car).
While we certainly want students in our education systems to be able to handle these life decisions, if we make no effort to educate them about mathematics itself, we rob everyone of an important opportunity. Mathematics tells the story of how the body/mind manages to keep looking, at what light cannot illuminate, trying to get past fragmentary perceptions and closer to the whole. It’s a fascinating story.
By Joselle, on September 12th, 2011 David Castelvecchi, at the Scientific American blog network, wrote about a Comment article that appeared in the July 13 issue of the journal Nature. The author, Peter Rowlett, takes note of what could happen when the mathematician “pushes ideas far into the abstract, well beyond where others would stop.” He does this with a collection of short pieces written by various authors about the unanticipated practical value of once purely theoretical considerations. I raise this point often in my own classes because I don’t believe these things are just happy accidents.
I agree with Castelvecchi that the piece seems to be motivated by funding issues, but it contains some nice stories. One of the more surprising references was the one about stacking oranges. Kepler (in 1611) conjectured that the way grocers stack oranges is the most efficient way to pack spheres. Thomas Hales finally proved this to be the case in 1998. It might seem that there is little value to the proof since we apparently already knew what we were proving. But proofs help build systems of thought about things that we don’t immediately understand. Packing problems are related to another kind of problem, one investigated by Newton in the seventeenth century, specifically: how many spheres can touch a given sphere with no overlaps? While Newton thought that in three dimensions the answer is 12, the proof of this was only given in 1953. It took until 2003 to find that the answer to this problem in four dimensions is 24. But the answer in five dimensions can still only be specified to be between 40 and 44. The eight dimensional version, however, was solved in 1979 when it was shown to be 240. This result is related again to the packing of spheres (in eight dimensions) now known as the E8 lattice. And it was the E8 lattice that was used in the 1970s for the development of a modem with 8-dimensional signals, opening up some new paths for internet signals to take.
One of the other examples in the Nature piece is one that is often on my mind, namely, topology. In 1735, Euler began thinking topologically when he demonstrated that it was not possible to find a walk through the city of Königsberg that would cross each of its bridges only once. Euler recognized that the only information needed to answer the question was the number of bridges and a list of their endpoints as represented in a schematic. The actual shape of the geography was irrelevant. It is the irrelevance of distance and shape in topology that I find most provocative. Rowlett gives a nice list of emergent applications:
Biologists learn knot theory to understand DNA. Computer scientists are using braids — intertwined strands of material running in the same direction — to build quantum computers, while colleagues down the corridor use the same theory to get robots moving. Engineers use one-sided Möbius strips to make more efficient conveyer belts. Doctors depend on homology theory to do brain scans, and cosmologists use it to understand how galaxies form. Mobile-phone companies use topology to identify the holes in network coverage; the phones themselves use topology to analyze the photos they take.
But what I find interesting about Castelvecchi’s blog is that he says this:
As much as I enjoyed the article, it must be said that picking some of the successful examples does not satisfactorily answer the broader question of whether the bulk of mathematical research is a “waste of time,” in the sense that it will never find applications anywhere. It is a legitimate question, and one that I am not qualified to answer.
I don’t think it is a legitimate question. I think it develops out of a shared misunderstanding about mathematics in particular, and about human nature in general. We assume to know too much about what we are doing or why. I don’t think mathematics is driven by the same kind of problem solving as the sciences. I think it often happens within the human organism’s drive to organize. Perhaps it is grounded in the nervous system’s need to process ongoing, overwhelming and complex sensory data, which is the first way that we know anything. And I believe that this is the very reason that the riddle of a physical problem might suddenly be solved by some independently constructed mathematical system. Mathematics is likely exploring organizational possibilities, and in this way extending the reach of cognitive processes. More importantly, there is no way to evaluate them if they are not associated with an application, and no way to anticipate how they might be. The idea that we are in a position to judge this fluid creativity (which is likely informed by things that are outside of our awareness) is foolhardy. It is like believing that one could direct the evolution of human culture.
By Joselle, on September 5th, 2011 Mark Turner, cognitive scientist at Case Western Reserve, wrote an article that was recently posted on the Social Science Research Network entitled The Embodied Mind and the Origins of Human Culture. He makes the point that our awareness is divorced from “Almost all the heavy lifting in human thought and action,” which is done “in the backstage of the mind.”
What we see in consciousness is not thought but the smallest tip of the iceberg— usually a simple, compressed product of thought, something to keep us going
He also points to what neuroscientists believe may be the number of synaptic connections in the brain, “about ten thousand times as many stars as astronomers think might be in the entire Milky Way galaxy.”
All those connections, inside your head, in a system weighing about 1.4 kilograms, working, working, working. The timing and phases of firing in neuronal groups, the suites of neuronal development in the brain, the electrochemical effect of neurotransmitters on receptors, the scope and mechanisms of neurobiological plasticity—all going on, in ways we cannot even begin to see directly.
Turner is moving toward a discussion of what has come to be called an embodied mind, a notion that relies on the idea that
Since brains are built to drive bodies, it is not a surprise that the nature of the body informs the nature of thought.
These ideas, among others, are used to contrast the current thinking in cognitive science with the now outdated idea (what Turner calls the formalist view) that “thought must be computational, and computation can be described formally…” While I can’t make a precise comparison, this does call to mind the failure of Hilbert’s program in mathematics. But to stress the uniquely human and non-computational aspect of cultural evolution, Turner looks at the cognitive side of surfing (on the ocean, not the web). I must admit that I was impatient with some of his surfing nostalgia, but then I found the point provocative and will likely make use of it again. Turner explains that in surfing:
Your movement is driven not by intention alone (“look where you want to go”) and not by responses to the environment alone, but by a blend of both: the wave gets to decide what it wants to do (an anthropomorphic blend), and you have to anticipate and respond, but you decide when and where you catch the wave and where you want to go. Every movement blends, almost instantaneously, all that physics and all those intentions. (italics my own) In walking or running, you are the motive force of moving along a path…Surfing isn’t just a cut-and-paste combination of things you already do. It is a blend of many of them, with startling emergent properties: in the blend, standing is a means of locomotion, and the way you stand is a means of changing your path.
What I find particularly interesting, and applicable to what we call intuition in mathematics, are the “startling emergent properties” that happen in a blend, unexpected things that are not present in the pieces that make up the blend, nor can they be found in the sum of those pieces. In a talk that Turner gave at a Fields Institute workshop on Cognitive Science and Mathematics, Turner discusses compressed blends, like the blend of directions and positions of boats on the water into the notion of a triangle. Once we have achieved the identity of a single triangle on the water, there comes an uncountable infinity of triangles fitting those constraints, but they are all packed into a single triangle. Blending and human memory integrate things that are not coherent and make use of memories that have become decoupled from our environment. And these may be the seeds of human culture or, for the purposes of this blog, of mathematical worlds. At the end of his talk, Turner offers the following from Philo of Alexandria,c. 20BCE – 40CE
How, then, was it likely that the mind of man being so small, contained in such small bulks as a brain or a heart, should have room for all the vastness of sky and universe, had it not been an inseparable portion of that divine and blessed soul? For no part of that which is divine cuts itself off and becomes separate, but does but extend itself. The mind, then, having obtained a share of the perfection which is in the whole, when it conceives of the universe, reaches out as widely as the bounds of the whole, and undergoes no severance; for its force is expansive.
After all of his imaginative thinking, Turner makes the most expected of remarks – something like, we hope to find a more rational explanation than God for the expansiveness of human cognition. I would argue that seeing that understanding stretches through the whole of the body, that it is not separate from life itself, would make Philo’s observation even more interesting. In particular, these words: “The mind, then, having obtained a share of the perfection which is in the whole…” Perhaps the word perfection is no longer fashionable, but we could imagine the mind, having a share of the completed (which is the whole), can reach out as widely as the bounds of the whole…..enjoying another ill-understood blend of physics and intention.
By Joselle, on August 29th, 2011 I have often said that I get particular pleasure from mathematics that defies common sense expectations. A simple example would be the observation that two things can be the same size even though one of them is contained in the other – like the set of natural numbers and the set of positive even integers. I enjoy these things, not because they suggest that there’s something really strange about mathematics, but rather because they suggest that my common sense is limited and, more importantly, I can get around it.
In a May, 2011 Scientific American blog, John Wilkins addresses the problem of how to reconcile the esoteric nature of science with Darwin’s evolutionary system based on fitness. He reminds us that even Darwin had some difficulty with this, reproducing part of a letter that Darwin wrote in 1881 where he says:
But then with me the horrid doubt always arises whether the convictions of man’s mind, which has been developed from the mind of the lower animals, are of any value or at all trustworthy. Would any one trust in the convictions of a monkey’s mind, if there are any convictions in such a mind?
And I think, why not? I continue to be surprised by what I want to call a prideful misrepresentation of the human world of ideas. The monkey’s mind is looking and giving structure to what it sees in the same way that ours is. It must be the very same perceiving and cognitive mechanisms whose actions are somehow extended and blossom into human culture.
I’ve collected a few references that point to work that may be identifying the springs, creeks, and small rivers through which we build our mental waterways:
Cells that may have helped with continuity:
It was recently reported that a study published in the journal Neuron shows that researchers have identified what they call ‘time cells,’ that is cells in the hippocampus that “robustly represented sequential memories…” Important to the study was the fact that time cell behavior connects related events that are separated by an interruption.
Each cell by itself provided a detailed ‘snapshot’ of the experience, and only at specific moments. But together, the activity from all of the cells filled in the gap,” said coauthor Dr. Christopher MacDonald. The appropriately named “time cells” that were active have much in common with previously described “place cells” that are active when animals are at particular locations in space. The time cells were able to adjust, or “retime,” when the duration of the delay period was altered.
In a summary of their work, the authors say the following:
The hippocampus is critical to remembering the flow of events in distinct experiences and, in doing so, bridges temporal gaps between discontiguous events. Here, we report a robust hippocampal representation of sequence memories, highlighted by “time cells” that encode successive moments during an empty temporal gap between the key events, while also encoding location and ongoing behavior. Furthermore, just as most place cells “remap” when a salient spatial cue is altered, most time cells form qualitatively different representations (“retime”) when the main temporal parameter is altered. Hippocampal neurons also differentially encode the key events and disambiguate different event sequences to compose unique, temporally organized representations of specific experiences. These findings suggest that hippocampal neural ensembles segment temporally organized memories much the same as they represent locations of important events in spatially defined environments.
It has often been discussed (although not so biologically) that our sense of time is a kind of template for the idea of continuity in mathematics, one that largely defines the character of analysis and dynamical systems (which are called flows when seen in real time).
Cells that create space and spatial coordinates:
This work on time cells says that they have much in common with previously discovered cells called place cells. The following is a discussion of place cells from Wikipedia:
Place cells show increased frequency of firing when an animal is in a specific area referred to as the cell’s place field…..When a rat forages randomly in an environment, place fields are only weakly modulated by the direction the rat faces, or not at all. However, when an animal engages in stereotyped behaviour (e.g. shuttling between goal locations), place cells tend to be active in the place field on passes in one direction only.
On initial exposure to a new environment, place fields become established within minutes….In a different environment, however, a cell may have a completely different place field or no place field at all. This phenomenon is referred to as “remapping”. In any particular environment, roughly 40-50% of the hippocampal place cells will be active.
In an environment in which a rat is constrained to walk along a linear track, place fields will often have a directional component in addition to a place component. A place cell that fires at a particular location while the rat walks in one direction along the track will not necessarily fire as the rat visits that location from the other direction.
But there’s more. There are grid cells that provide geometric coordinates for location, generating an internal grid to help with navigation. There are also head direction cells that act like a compass. You can find recent work on grid cells here and a more recent one here. There’s also a youtube video on place cells.
It’s easier to talk about what is often called our accidental number sense (an innate ability to distinguish one from two or three things, or large collections from significantly smaller ones) when we consider intuition in mathematics. And this is probably because, like mathematics, we can see our use of number sense in our conscious mind. But, for me, these cellular level organizations of sensory data are much more provocative and better suited to the idea that we must be using some less than conscious part of us to explore mathematical possibilities.
By Joselle, on August 22nd, 2011 A number of websites have reported on a recent study, that correlated innate number sense with mathematical ability. A concise report of the study can be found in the Johns Hopkins University Gazette, published by the institution where the study was done. The study’s results confirm a correlation between the strength of ones number sense and the development of other mathematical abilities. New to this finding, however, was that participants in the study were young children. None of them had yet had the formal schooling that could obscure the correlation or confuse its interpretation. The article entitled, You Can Count On This, says clearly:
According to the researchers, this means that inborn numerical estimation abilities are linked to achievement (or lack thereof) in school mathematics.
A number of things went through my head as I read the report (and responses to it) and I would like to say something about a few of them.
First, I think it is dangerous to express these observations in a way that suggests that one can be born with or without mathematical ability. One of the reports I saw on the study took the results in exactly this direction. The article was called People Are Born Good or Bad at Maths and says this:
Now the researchers, who carried out tests on children too young to have been taught mathematics, found that people are either born with a mathematical brain or not, news paper reported.
This is not a view that should be encouraged. And there is good reason to be reserved about the correlation observed. Mathematics is expressed with number, but it is not defined by number. Number sense is not the beginning of mathematics (and hence mathematical ability). It is true that one of our earliest expressions of mathematics happened in the symbolic representation of quantity. But, as we all know, this was not its only early expression. The other was spatial, and took the form of geometry. Only with the persistent mingling of these, could we profit from the vitality we find in every instance of their overlap. This is the energy that brings life to modern mathematics.
It is more likely that mathematics rests on manifold, interwoven, perceiving mechanisms. My hunch is that these are actually reflected in the reasoning that builds mathematical structure. It does seem that the idea of equivalence begins with counting, but equivalence is found in spatial relationships as well (like the simple idea of congruence) and mathematics’ many powerful extensions of equivalence would not be possible without, at least, these two notions.
Mathematics is challenging, sometimes very difficult. Some problems will even seem impossible. It should not be made easier for students, at various levels of instruction, to decide that they’re just not built for it. A teacher of mine once said that doing mathematics relies on ones willingness to look stupid. We don’t understand enough about ourselves, or even about what mathematics is, to think too broadly about a correlation between number sense and some facility with numerical ideas. I like the words Richard Courant chooses to describe the basic elements of mathematics (in the well-known What is Mathematics?)
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
I also want to draw attention to a NY Times report on a related study from a few years ago. This article was given a different kind of title. It’s called Gut Instinct’s Surprising Role in Math and describes a reservation on the part of investigators:
The researchers caution that they have no idea yet how the two number systems interact. Brain imaging studies have traced the approximate number sense to a specific neural structure called the intraparietal sulcus, which also helps assess features like an object’s magnitude and distance. Symbolic math, by contrast, operates along a more widely distributed circuitry, activating many of the prefrontal regions of the brain that we associate with being human. Somewhere, local and global must be hooked up to a party line.
In his book The Lightness of Being, Frank Wilczek compares the expressions of Einstein’s first and second law. His point is that “different ways of writing the same equation can suggest very different things, even if they are logically equivalent.”
Einstein’s first law is: E = mc² His second law is: m=E/c²
The first one, Wilczek argues, suggests the possibility of getting large amounts of energy from small amounts of mass (images of bombs and nuclear power plants). The second law, however, suggests that we might be able to understand how mass arises from energy, or that we may have some handle on the creation of mass (which has the attention of high energy experiments going on right now).
In this light, I’d like to end this with another quote from that NY Times article from a few years ago, where researchers are looking at the same pieces of information and seeing something slightly different. They say:
What’s interesting and surprising in our results is that the same system we spend years trying to acquire in school, and that we use to send a man to the moon, and that has inspired the likes of Plato, Einstein and Stephen Hawking, has something in common with what a rat is doing when it’s out hunting for food. I find that deeply moving.
By Joselle, on August 15th, 2011 I recently listened to a radiolab podcast (from this past November!) that featured two authors: Steven Johnson (author of Where Good Ideas Come From) and Kevin Kelly (author of What Technology Wants). The thrust of the argument, that both authors defended, was that the things we make (from tools to gadgets to computers) are an extension of the evolutionary processes that make us. There is a lot to be gained from this perspective. However, the proposal that there is an individual-like willfulness in the technology, may go a bit too far. I say this not because the technology is seen as living (which I think it is) but because we understand so little about the source and nature of our own willfulness. Yet I found myself willing to allow it. Perhaps because an extreme like this is necessary to nudge the common perspective (which I think is wrong) away from the idea that individuals are discreet entities, fully aware of themselves, what they want, and what they are doing. The biological relatedness of ourselves and our equipment is very interesting and very plausible but we need to understand ‘the will’ better before we begin extending it to our equipment.
I’m more interested, however, in the fundamental idea that brings Steven Johnson to the table, and is very nicely rendered by him. His talk at a TED conference summarizes his perspective. There he says the following:
…an idea is a network on the most elemental level. I mean, this is what is happening inside your brain. A new idea is a new network of neurons firing in sync with each other inside your brain. It’s a new configuration that has never formed before. And the question is: how do you get your brain into environments where these new networks are going to be more likely to form? And it turns out that, in fact, the kind of network patterns of the outside world mimic a lot of the network patterns of the internal world of the human brain.
This sense that interior worlds and exterior ones mirror each other will inevitably provoke some useful new philosophical thoughts. And there are two other metaphors Johnson uses that I like very much because they match up with my own sense about things, including mathematics. They are: the slow hunch and the liquid network. In his book he refers to the concept of flow, that was coined by psychologist Mihaly Csikszentmihalyi, to describe the internal state of a productive mind. He says:
It’s a lovely metaphor precisely because it suggests the essential fluidity that good ideas often need……it is not the miraculous illumination of a sudden brainstorm. Rather, it is more the feeling of drifting along a stream, being carried in a clear direction, but still tossed in surprising ways by the eddies and whirls of moving water.
I particularly like this because it is about water, encouraging one to see ‘the thought’ in ‘the flesh’ again. It can also be used, quite easily, to characterize the development of ideas in mathematics’ history.
The slow hunch lines up with two things I like to think about. One, is the centuries of introspective labor that move mathematics forward. But the other is illustrated in the story he tells about Darwin. Johnson retells Darwin’s own account of how the basic algorithm of natural selection “kind of pops into his head.” But, Howard Gruber, who went back and looked at Darwin’s notebooks from the time, found that Darwin had the full theory, with apparent textbook clarity, for months and months before “his alleged epiphany.” What Johnson proposes is that Darwin had the idea but in some way was unable to fully think it. This not only illustrates the nature of “the slow hunch” but it demonstrates the spottiness of our awareness, the way our own thoughts can produce something that we don’t actually comprehend yet. This is a way that I have thought about mathematics when I’ve looked at its many conceptually extended notions – like the various extensions of our original number concept till they lead to the real number continuum (or the value of the complex number for that matter). I continue to think that the earlier notions are not built up into these extensions, but are instead already part of the larger ones when we first glimpse them. It may not be easy to defend this view, but trends in cognitive science, neuroscience, culture studies, even evolutionary psychology usually help me support it.
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