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By Joselle, on September 4th, 2010 As a brief follow-up to my last post, I reread some pages in Tobias Dantzig’s book Number and, on one of them, he is critical of early 20th century formalists when he says about symbols in mathematics:
To me the tremendous importance of this symbolism lies not in these sterile attempts to banish intuition from the realm of human thought, but in its unlimited power to aid intuition in creating new forms of thought.
The question I wanted to raise in my last post is this. Perhaps symbol has flourished so well in mathematics that our thorough comprehension of them is actually lagging. That we can’t quite see some of the windows they open up to our intuition.
By Joselle, on August 30th, 2010 I often get stuck in the gap we rarely notice between the word and what the word is meant to signify. Do we really understand what word and symbol actually do? Or, even more to the point, can we see what the body is accomplishing in the evolution of these cognitive tools?
In a film I saw recently about Alaska we were told that a people native to the area once said that the northern lights were the spirits of children before they were born. The narrator of the film went on to describe our understanding of the lights. He began with the phrase, “scientists say the northern lights are…..” In my own mind, I saw this physical account of the lights as an organization of ideas (of the words photons, atoms, solar wind, etc. and their associated meanings). I saw it as the modern naming of what we see, given the depth and precision physics has brought to the senses. I wasn’t challenging this description, just taking note of its reliance on our scientifically refined analysis of sensory information.
I had a similar experience when I read the article which motivated this post. The article, Back From the Future, was published online by Discover on August 26 and it is a discussion of research in quantum mechanics designed to establish that information can flow back from the future, i.e., that in quantum mechanics, the arrow of time is symmetric. There are so many ordered thoughts here…time, information, arrows, direction, flow, let alone the complexity of the mathematics of quantum mechanics.
It seems the term arrow of time made an early appearance in 1928 in the book The Nature of the Physical World. The author, British astronomer Arthur Eddington states:
Let us draw an arrow arbitrarily. If as we follow the arrow we find more and more of the random element in the state of the world, then the arrow is pointing towards the future; if the random element decreases the arrow points towards the past. That is the only distinction known to physics. This follows at once if our fundamental contention is admitted that the introduction of randomness is the only thing which cannot be undone. I shall use the phrase ‘time’s arrow’ to express this one-way property of time which has no analogue in space.
The emphasis is mine.
The Discover article describes how quantum mechanical ideas have confounded some of our earlier views of the physical world and, in particular, how they altered the deterministic view of things. The work described is driven, at least in part, by a desire to save a deterministic view of the universe. About this new model of quantum mechanics, the article says:
It could produce all the same treats as the standard form of quantum mechanics that everyone knew and loved, with the added benefit of explaining how information from the future could fill in the indeterministic gaps in the present. But while many of Aharonov’s colleagues conceded that the idea was built on elegant mathematics, its philosophical implications were hard to swallow.
An article in the Stanford Encyclopedia of Philosophy highlights the role of mathematics in quantum mechanics:
Quantum mechanics is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic particles — or, at least, of the measuring instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any theory we have ever had. Mathematically, the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to quantum mechanics.
Our dependence on mathematics in physics seems to widen the gap between the words (or symbols) and what they signify. The words used in a description of the northern lights are much more consistent with more common sensory experience. But the mathematics of quantum mechanics and the physical concepts these relationships generate are bewildering, despite our confidence that they do reflect the nature of the world around us. Maybe the evolution of word and symbol, or their emergent growth, has exceeded our understanding of how they operate in us, what they bring or, more specifically, what the body accomplishes when it creates these representations. And this leads to the odd sensation that we can’t quite comprehend what our own symbols are telling us. While this may seem to not make sense, given that so much of the brain’s activity happens outside of our awareness, perhaps it can be considered.
By Joselle, on August 23rd, 2010 Most of us begin drawings with lines. And even though those lines may not be in the subject of a rendering, they are nonetheless perceived. Some of the visual information we use to re-present our experience in a drawing is also used in mathematics, geometry in particular. A difficult question to answer but an interesting one to ask is, “What is it that allows (or inspires) us to isolate some of the visual attributes of an object (like straight or curved lines or closed figures)?”
My father taught me how to draw when I was a child. He was very good at it and, as far as I knew, he had no training. “Draw what you see, not what you know,” was his consistent warning to me. And today I saw this phrase echoed in a number of web sites. I also found a paper from members of the Department of Psychology at the University of North Carolina, Wilmington. (The pdf can be found here) that explores my dad’s admonition. The paper is more than ten years old, but its authors set up experiments in order to isolate the major reason most people find drawing difficult. They concluded that things like motor coordination, the artist’s choice about what aspect of an object to represent or the artist’s misperception the drawing’s accuracy, had very little effect on the success of the drawing. Instead, it was decided that the artist’s misperception of the object itself was the problem.
The paper sites two ways we misperceive – illusions and delusions. But since “both poor and accomplished artists are affected by illusions,” these were not investigated with experimental trials. (It is worth noting, however, the current work on understanding illusions ). The researchers found delusions, defined as the memorized ideal of an object, to be the most significant factor in drawing inaccuracies. They describe earlier work on delusions with children:
When copying an outline drawing of a table, for example, children make systematic errors that correspond to their knowledge of what a table looks like. However, when children copy outline drawings of parts of the table in isolation, they make very few copying errors (Lee, 1989). These results indicate that the children’s knowledge of the form of a table is interfering with the accuracy of their drawings.
I think this captures the significance of “draw what you see, not what you know.” But the phrase “memorized ideal of an object” deserves some attention. Idealizing seems to be part of our nature. As if a visual ideal, like the table, might invite us to consider other ideals, say a mathematical ideal, like a circle, or a philosophical ideal like truth.
But our memory is not like a collection electrochemical versions of photos that we cut and paste. Memories seem to depend on the complex interactions of many brain processes. A study was done to record the brain’s activity during the process of drawing faces. Participants were asked to reproduce black and white cartoons of faces. The interesting thing:
The results show that looking at the cartoons activated visual processing areas of the brain, that are known to be responsive to faces, especially if the cartoon was displayed at the same time as they produced the drawing. But when the subjects had to wait before drawing, there was no maintained activity in these areas. This suggests that the memory of the cartoon face is transformed into a different, non-visual form.
The non-visual form is handled by the brain as spatial information.
They conclude that facial information is captured during a sequence of eye movements towards certain features of the cartoons, and the information is stored as spatial locations for subsequent eye and hand actions.
Now back to mathematics one more time. Geometries use visual information and a geometric understanding of an analytic idea will ‘visualize’ the idea (the complex plane for example). But the generalities growing out of geometric ideas (n-dimensional spaces for example) transgress the three-dimensional limit of our visual imagination. Perhaps the strength of their mathematical relations, and their reliability, rests, at least in part, on the way these mimic interacting brain processes that we can’t quite see yet.
As to why visual attributes are isolated and explored in the first place, I believe, like Semir Zeki, that the body is looking for the essence of things.
By Joselle, on August 16th, 2010 I just heard Radiolab’s show on words (you can listen here). The show explores just how much of our experience is born of language. It begins with experiments which seem to reveal that until we can bridge islands of our experience with phrases, we can’t actually think. This may be a difficult argument to make since what we mean by ‘thinking’ may just be bridging experiences with words. But the difficulty one might have articulating exactly what experiments are showing us, just highlights how fascinating the topic really is.
In usual radiolab fashion, we hear about rats that don’t think, then about a 27 year old deaf man who was without language until Susan Schaller found a way to teach him, followed by some talk from a Shakespeare scholar who has unexpected information about words invented by Shakespeare. We also hear Jill Bolte Taylor talk about her own stroke (the subject of her book My Stroke of Insight). And we hear the story of 50 deaf children who, when brought into community, developed their own language.
A central idea in the broadcast is that words themselves, brought together in different combinations (infinite in number), actually create new vision – even something as fundamental as the perception of one’s self. For me, this touches on the development and effectiveness of mathematics, as it names perceived objects (even if they are perceived internally) and brings them into relations with each other, extending what we see yet again.
What do you think?
By Joselle, on August 11th, 2010 The story of online gamers solving protein structure problems for biochemists has been reported by many, including the New York Times, NPR’s 360, Youtube, and a host of blogs. The gamers, by using their three-dimensional puzzle solving skills, have made significant contributions to biochemical research.
The problem for biochemists is predicting the shape that a protein will take. Proteins first appear as strings of amino acids, like coils. The amino acids then interact with each other to produce a well-defined three-dimensional structure. It happens spontaneously and quickly, in time frames measured in billionths of a second. The process that takes the protein from string or coil to the three dimensional structure is what’s called the folding. And even if the time frames were slower, proteins are far too small to be seen by a microscope.
Researchers at the University of Washington became aware of inefficiencies in the software that had been designed to find the folding patterns. They considered that one of the problems was that the computer couldn’t see the shapes. But individuals could see the inefficiencies, so perhaps individuals could also work through the complexities of the whole problem. And so they designed a a protein-folding video game, available on the Internet, to tempt the human imagination to solve the puzzles. The researchers themselves had no advantage in solving the puzzles. In fact, they weren’t good at it. The game, called Foldit, has attracted a dedicated following of thousands of players (one who spoke on NPR was only 13). Players actually fold different proteins into their lowest energy state (which is their final resting state) using computer tools and figures that are designed to contain the biochemical parameters of the problem. The success of the effort was reported in a paper in the journal Nature, and the players were listed as coauthors of the paper.
The complexity of the problem is directly related to the degrees of freedom, or the number of parameters, in the space of all possible shapes. But the Nature paper tells us that one of the reasons the human effort surpasses the software effort was that gamers (who are able to collaborate) didn’t only look at all possible shapes. They also looked at all possible strategies.
The times article tells us that, according to one of the gamers, the complexity of a problem is somehow like “……trying to solve a million-sided Rubik’s Cube while it spins at 10,000 r.p.m.” Also, according to the Times:
In a comparison involving 10 separate protein-folding puzzles, video game players matched the results generated by software solutions in three of the puzzles, outperformed them in five cases and found significantly better solutions in two others, according to the scientists.
Knowing the shape of particular proteins is the key to understanding how they work (even as they cause the progression of a disease like HIV).
This is certainly a new way to do science and opens the door, I’m sure, to all kinds of ways that the human community can be knitted together. But it’s also worth noting that the success of Foldit says something about the talents of the imagination. We’re wired for pattern recognition and for conceptualizing problems, and this is the very thing that makes mathematics possible. That gamers explored the strategy ‘space’ as well as the structural ‘space’ is a great example of the determination of our imagination. I’ve also learned that dopamine can make the puzzle irrisistable since its pleasurable release is now known to happen, not when we finish a puzzle, but when we start it. But that’s another story that I’ll write about soon.
By Joselle, on August 9th, 2010 I just heard a story on NPR’s All Things Considered that centered on when we became mentally modern human beings. In our evolutionary history, the show argues, the appearance of symbolic thought marks the genesis of uniquely human developments more than say, standing upright. And I agree. On a daily basis, we move more from idea to idea than from place to place.
One of the surprises in the story was that when a few people were asked the question “What is symbolic thought?” they didn’t think they knew. When asked about what a symbol is, they got stuck on fairly big individual symbols like flags, peace symbols and crosses. They didn’t consider, for example, the word tree or the number 2. In fact, in a show about symbolic thought, mathematics didn’t come up at all.
Notches cut into a baboon bone that is approximately 35,000 years old is evidence of some of the earliest mental organizing we did. Counting, it seems, predates language. And the age of the notched baboon bone corresponds, in time, to some of the earliest cave paintings found. The mind is exploring so much in those paintings and expressing what it finds. These are clearly very early demonstrations of our talent for abstraction or symbolic thought and they are the ground that grows the vast conceptual landscapes in which we live today.
But the NPR story wanted to push things even further back, and found what they called a fossil record of early symbolic thought in shells collected by archeologist Chris Henshilwood. The shells were found in sand layers from 75,000 years ago on the western coast of South Africa. They had little holes in them, that came from being strung, like beads. Henshilwood is convinced that the beads had meaning, that they said something about the person wearing them, i.e.,that they were symbolic. We don’t usually think of jewelry as symbolic thinking. But there is meaning in much of it, the wedding band, worn on the left hand, or the many symbolic things we wear around our necks.
Worth taking note of in this story is the difficulty we have actually seeing ourselves (hence the blank minds when asked about symbolic thought) and the unexpected ways we get a glimpse (like shells in 75,000 year old sand). Taking a fresh look, without thinking we already know, is the way to valuable insight. And I’m committed to the idea that a fresh look at what mathematics does, without thinking we already know what it is, will get us a pretty interesting look at ourselves.
By Joselle, on August 4th, 2010 I just read through a series of blogs generated by an article in the journal Neurosurgery, in which two neurosurgery researchers at Johns Hopkins University argue that an anatomically accurate image of the human brain is hidden in God’s neck in one of Michelangelo’s frescos. I was struck by how little the reports and blogs had to say about it, which makes some sense considering we are looking at a medical/scientific image in a painting from the 16th century, by an artist whose name identifies the work and the time more than the man. There’s something incongruent about all of it. What Michelangelo may have actually been thinking seems way out of our reach. But I wanted to read some interesting things about what it could mean none-the-less. I wanted to think about it with somebody. There was, of course, one art historian, quoted in many of the reports, who thought the whole thing was nonsense.
I’m often frustrated by how compartmentalized our thinking has become. The content of the various disciplines contained in the words Arts and Sciences has grown so vast, that accomplishment requires specialization. And then there’s religion. But all of these things are built from some one world by virtue of the fact that they are all human. Certainly specialization is an effective tool. It is the specialization of cells that makes our existence possible. But our experience is a consequence of the extent to which they can be coordinated. The brain is thought to have specialized areas but much of the character of a human life is made from how they’re shared – how, for example, visual functions are extended to thought functions, mental images and forms, language and even mathematics.
Michelangelo was one man doing several things and doing them extraordinarily well. He did have a rocky relationship with the Church. But he lived in a time when art, science and religion had not developed the clear boundaries that separate them today. Why would someone of his stature, in his 16th century world, hide an anatomical image in God’s neck?. Is it hidden because the Church disapproved of the dissection of cadavers being done by students of anatomy? And he put it there like a rebellious child? Or is it part of the story he’s telling in the paintings themselves? Does it have some symbolic meaning? The brain stem is fundamental to human experience and thought. What does it mean to have it in what Douglas Field’s article calls the voice-box of God? Could it be an expression of the fact that God and the flesh are related, or some material expression of the Christian idea of God within us, or of how we may hear God? ‘The flesh,’ after all is given a lot of attention in the Gospels.
Dr. Field’s article for the Huffington Post (also showing as a guest blog at Scientific American) was the only post I enjoyed.
I’ll end with this quote from it:
Perhaps the meaning in the Sistine Chapel is not of God giving intelligence to Adam, but rather that intelligence and observation — and the bodily organ that makes them possible –lead, without the necessity of Church, directly to God. The material is rich for speculation and the new findings will doubtlessly spark endless interpretation. We may never know the truth, but in Separation of Light from Darkness, Michelangelo’s masterpiece combines the worlds of art, religion, science, and faith in a provocative and awe inspiring work of art, which may also be a mirror.
By Joselle, on June 25th, 2010 This may not be a timely commentary, but I only recently read the book Naming Infinity (Harvard University Press 2009). It was a gift from my husband who rightly expected that I would be interested in a book purported to be about how mathematicians were supported through a conceptual crisis by the bold work of believers in the mystical tradition of name worship. The authors can never fully display the correspondence between the belief in name worshipping and the mathematics itself, but they do successfully tell the story of the way the religious, emotional, mathematical and political lives of a group of Russian mathematicians converged. This is certainly a story worth telling. The passion and devotion we are often lead to see in stories about artists or saints is seen here in these mathematicians’ lives. Just this glimpse of their dedication is enough to tell us that something more is happening in mathematics than mere problem solving.
There have been a number of disputes among mathematicians, particularly through the late 19th century and early 20th centuries, about which math ideas were legitimate and which ones should not be allowed in the discipline. Interestingly enough, along with set theory notions and characterizations of the infinite, there was even objection to discontinuous functions. Reading the different positions in these arguments shows us something about the nature of mathematics itself and the intellectual and psychological struggles it can create. This particular book highlights an important question, namely – what does it mean for a mathematical object to exist? How does naming something contribute to or even produce its existence? These are beautiful questions and the answers are not obvious. I hope to continue to discuss them in upcoming blogs.
The Grahm/Kantor book does a very nice job of revealing history that will surprise us and it brings us into the world of some Russian mathematicians we may know little about.
By Joselle, on January 9th, 2010 Advances in neurobiological research often demonstrate how very difficult it is for us to get a good look at ourselves. Effective analytic tools in the sciences usually rely on defined categories such as organic and inorganic; animals and plants; protons, neutrons and electrons; voluntary action versus involuntary action; motor skills versus thinking skills. But refinements in our analytic efforts often require redefining the very categories they have created. The famous particle/wave duality of light is a perfect example.
A recent article about a particular brain structure contributed to my fascination with how thought-governed lives emerge from the basic aspects of our biology that we share with so many creatures. The article is called Seeing Without Looking: Brain Structure Crucial for Moving the Mind’s Spotlight. It summarizes findings reported in a December issue of Nature. It considers the relationship between looking at something with your eyes and paying attention to something with, for lack of a better word, your mind.
The brain structure in question is called the superior colliculus. Its function has been understood to be the motor control of head and eye movement, i.e. sending motor control commands to eye and neck muscles. But experiments at the Salk Institute for Biological Studies indicate that the superior colliculus is equally involved when you move your attention away from the thing you may be looking at. And the institute, it seems, has been paying attention to the superior colliculus for some time. Another study was reported on in September 2008.
This one observed what happens when we track what researchers called “the invisible center” of a moving object. The invisible center of something is like an airplane whose presence at night can only inferred from peripheral lights, say on its wings. Since the superior colliculus contains a topographic map of the visual space around us, it mirrors geographic space, and it is possible to identify neurons that correspond to the spot in this space where our eyes would focus (named the foveal location). The studies confirmed that neurons in the foveal location are active even when the object of their attention was invisible, like the dark airplane. The eyes seem to be pointing to the invisible part of the image.
In February 2009, researchers at Salk also found that the superior colliculus controls microsaccades, those quick tiny eye-movements necessary to keep visual images from fading and that appear to be random. But according to Richard Krauzlis, an associate professor in the Salk laboratory “…results show that the neural circuit for generating microsaccades is essentially the same as that for voluntary eye movements. This implies that they are caused by the minute fluctuations in how the brain represents where you want to look.” (emphasis mine). It was demonstrated that even if we avert our eyes away from an object that gets our attention, the direction of microsaccades will be biased toward that object.
These observations suggest an interesting link between our eyes and the more general action of just paying attention to something and thus also indicate some overlap of reflexive (automatic) action and thoughtful (deliberative) behavior. Voluntary and involuntary movements share neurons. It was also noted in the December 2009 article, (again quoting Richard Krauzlis) “… results show that deciding what to attend to and what to ignore is not just accomplished with the neocortex and thalamus, but also depends on phylogenetically older structures in the brainstem.”
When a brain structure that seems built to move eye and neck muscles is also found active in moving our attention, purely mindful attention becomes linked to sight. That the eyes will lock on an invisible object suggests that something of the visual brain will respond to internal stimuli. And finding that neural circuits for what appear to be reflexive eye movement, namely microsaccades, are essentially the same as the ones for voluntary movement, suggests that distinguishing between voluntary and involuntary is not so easy given that microsaccades happen completely outside of our awareness.
Each of these studies focusses on a detail that would seem to have no affect on how we see ourselves. But they contribute to the steady progress neuruoscientists are making as they try to unravel what the brain is doing. This unraveling often leads to novel considerations like Semir Zeki’s idea that the visual arts are an extension of the function of the visual brain (August 2009 post).
The body is built to be in its world, to see it and move through it and use it. Our elaborate conceptual structures built with language, reason, mathematics and all of our scientific efforts are inevitably grounded in fundamental biological actions and may be motivated by more than our awareness can discern. It is entirely reasonable to consider that we can never fully understand what we’re doing or why.
By Joselle, on November 14th, 2009 It has been understood for some time that metaphor provides a sensory anchor to abstract ideas. But, more recently, cognitive psychologists have looked at how active the role of metaphor may be in thinking. In a recent article on Boston.com, experiments are cited which explore the extent to which metaphor shapes thought.
The article cites studies where subjects were given a cup of hot or cold coffee to hold, without being told that it was part of the study and, a few minutes later, they were asked to characterize a person that was described to them. The subjects were more likely to find that person to be caring, generous, or good-natured if they were holding a warm coffee than if they were holding an iced one. In another study, participants were less likely to describe a social situation as having gone smoothly if they had handled some sandpaper covered puzzle pieces. It all sounds a bit unreasonable, but that may just be because we underestimate the extent to which our thoughts rise out of our bodies.
The article also refers to two books by George Lakoff and Mark Johnson which have metaphor as the very root of thought (instead of what we might otherwise think, which is that the metaphor develops to clarify the thought). Lakoff uses this idea to do a cognitive study of mathematics in his book Where Mathematics Comes From, where even the most sophisticated mathematical concepts are understood to be grounded in sensation.
It’s difficult to grapple with the notion that the body someow leads in the building of ideas or that we are never fully aware of the source of our thoughts. But I have long thought it clear that the obscurity of mathematics’ source and its surprising breadth of understanding is a signal that we don’t completely understand how thoughts happen or what they may be accomplishing. Results in mathematics are often unanticipated and it frequently happens that a concept finds application long after it was developed. Giving the body its due may help dampen our inclination to be smug about what is known, or correct the complicated conflicts produced by certainties evidenced by social division and injustice.
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