I’m planning to post something new this week, but I would also like to share the link to my guest blog for Scientific American that was posted last week. The title of the piece is: To What Extent Do We See With Mathematics?
Hope you enjoy it.
I’m planning to post something new this week, but I would also like to share the link to my guest blog for Scientific American that was posted last week. The title of the piece is: To What Extent Do We See With Mathematics?
Hope you enjoy it.
Much of the research done in cognitive science is designed to study the development of concepts – internal representations that define the idea-driven nature of modern human experience. And, in our experience, it’s difficult to mend the rift that’s been created between what we call thought and what we call reality. But a number of paths have opened up within various ‘embodied mind’ theses that are intended to correct the mistaken duality. I’ve covered some of them in previous blogs. I’ll refer here, again, to biologist and philosopher Humberto Maturana, a proponent of a closely related idea known as enactivism. In a paper that appeared in Cybernetics and Human Knowing in 2002, Maturana refuted the idea that language is a collection of abstract representations that correspond to concrete things, and made the following observation:
Languaging is action. And this is how I have come to see mathematics, as action.
To verify that the stone was a sundial, archeologist Larisa Vodolazhskaya calculated the angles that would have been created by the sun and shadows at that latitude. She confirmed that the carvings on the slabs marked the hours accurately. “They are made for the geographic latitude at which the sundials were found,” she said. And these ancient carvings rely on a sophisticated grasp of geometry.
The circular depressions, placed in an elliptical pattern, are hour marks of an analemmatic sundial; the largest groove on the plate, Vodolazhskaya said, marks where the vertical, shadow-casting gnomon would have been placed at the winter solstice.
Meanwhile, a long carved line transected by a number of parallel grooves in the center of the slab would have acted as a linear scale for a more traditional horizontal sundial, where the hours are marked by a gnomon’s shadow falling along hour lines. In this case, the horizontal sundial actually had two gnomons, Vodolazhskaya said. One gnomon tracked the time in the morning hours and early afternoon, and the second covered from late morning to evening, measuring time in half-hour increments. Ancient sundials with half-hour marks are rare, though one was discovered earlier this year at the Valley of the Kings in Egypt.
A thorough analysis of the stone can be found in a paper by Vodolazhskaya. The figures in the paper illustrate the impressive precision of the stone’s markings. The sophistication of the geometry employed and the sundial’s ability to measure the passage of time in half hour increments are taken as evidence that ancient Egypt had some influence on the people who inhabited the northern coast of the Black Sea. What strikes me, however, is something that may be unique to an ancient ‘tool’ like this one. In some sense, the intricate geometric structure exhibited by the stone exists only when the stone and the sun are taken together. It emerges from the directed light of the sun together with human perception, thought, and craft. This would be true of any ancient sundial. It just happens that this one made me particularly aware of it, perhaps because of the fine tuning between this one and the latitude of the location where is was found. In the right light, the mathematics isn’t fully isolated from the action of the sun and the sundial maker.
The mathematics isn’t describing the relationship between sunlight, shadow and time, it is the relationship between sunlight, shadow and time. And this is why I thought again about Humberto Maturanna. I think that mathematics, like language, is a kind of shared action, occurring within and among things in the world. It exists in various bodies that may or may not find reason to express it formally.
If you haven’t seen it, the more self-contained mathematical action in the little film Nature by Numbers is worth a look.
The Atlantic Monthly just did an interesting piece on Douglas Hofstadter, Pulitzer Prize-winning author of Gödel, Bach and Escher. Hofstadter’s 1979 book investigates the nature of human thought processes by looking at common themes in the work of the mathematician Gödel, the musician Bach and the artist Escher. In particular, it addresses the question of how meaning is born from apparently ‘meaningless’ elements – like the fundamental biological materials of the body and the brain, or the fundamental symbols of mathematics.
While once seen as groundbreaking work in artificial intelligence, the book is not about how to mechanically replicate human thought, but more about what thought ‘is,’ and the emergence of ‘meaning.’ MIT has offered a course on the text for high school students. Lecture videos and lecture notes for these courses can be found here. Early in one of the lectures, the point is made that the premise of Gödel, Bach and Escher is to show how the “I” of our experience, or the event of self-referencing in our experience, corresponds to the the self referencing that happens within mathematics.
James Somers, author of the Atlantic Monthly piece, focuses on an important shift in how Hofstadter’s work is understood.
The contrast between the progression of AI efforts and the aim of Hofstadter’s work is particularly relevant to my interests. Hofstadter’s persistence can be seen in the research of what was once called the Fluid Analogies Research Group or FARG at Indiana University. The group has been renamed The Center for Research on Concepts and Cognition. At the end of their research overview, they make the following statement:
This philosophy is usually traced back to a computer program that Hofstadter built called Jumbo. The program was inspired by his interest in understanding what was happening when one solved a newspaper jumble. He had no interest in the program that would quickly solve them. He wanted to understand what was happening when we solved them. As Somers points out:
The key for me is the efficacy of what is here referred to as “subcognitive” acts, acts that have some randomness about them, and may even work at cross-purposes. Hofstadter’s philosophy is also evident in his vast collection of speech errors, or his record of instances of swapped syllables, like “hypodeemic nerdle.” Again Somers points out:
The views on ‘thinking’ explored at the research center are provocative and carry large numbers of implications that will likely impact the age-old mind/body debate. They imagine ‘meaningfulness’ within the life of the whole of nature, of which mathematics is very much a part. I plan to spend a lot more time looking at work being done at the Center for Research on Concepts and Cognition and hopefully more time writing about it.
New Scientist published an article by Amanda Gefter in their August 15 issue which describes how and why the notion of infinity has come into question again. The distinction between a potential infinity (the process of something happening without end), and an actual infinity (represented, for example, by the set of real numbers) was disputed among mathematicians for a long time until Cantor brought new meaning to the nature of infinities in the set theory he created. Now infinities are part of the living tissue of mathematics. But infinities have thwarted the success of some physical theories and continue to do so. The survey of challenges from mathematicians and physicists alike make for an interesting read. One of the more reasonable claims comes from Max Tegmark of MIT.
I believe Riemann himself had not decided whether space was continuous or discrete.
One of the links within this piece was to another New Scientist article by Gefter, published in October of last year. The older piece had the title: Reality: Is everything made of numbers? In it, Amanda Gefter surveys some of emerging perspectives in physics that equate, in one way or another, mathematical reality with physical reality. She recalls Einstein’s fix for the equations that described an expanding universe, years before there was clear evidence that the equations were correct.
With respect to current physics investigations she says:
Gefter then tells us about how some prominent physicists, like Brian Greene and Max Tegmark, are tackling the riddle. She has more to say about Tegmark’s view than Greene’s, and Tegmark’s view is fairly extreme. Gefter quotes him:
I’ve always liked this kind of talk. While the content of a remark like this might be difficult to specify, I suspect that it’s full of insight. Tegmark goes on to suggest that the mathematical structures that have no physical application in this universe correspond to other universes. But his ideas rest, in part, on his judgment that mathematical structures don’t exist in space and time. ” Space and time themselves,” he says, “are contained within larger mathematical structures.”
Now, one might argue that brains and computers exist in space and time, and without them there is no mathematics, or at least not the kind we used to thinking about. But I don’t want to move here into a difficult philosophical debate. Instead, I’d like to suggest that if we consider that everything that exists is known only in relationship, some new light might be shed on the question of how mathematics can do what it does. Physicists like Tegmark can seem to be replacing the physical subjects of their investigations with mathematical ones, putting the mathematics ‘out there.’ Cognitive scientists seem to be finding mathematical notions mirrored in sensory and learning processes, suggesting that mathematics is part of how the body learns about its world. But the body’s perceiving mechanisms are built entirely in relation to its surroundings, or more specifically, to the properties of things like air and light, or even gravity. If we can see mathematics as one of the body’s actions, action that stretches the reach of its perceiving and learning mechanisms, then perhaps we can imagine that it develops not ‘for’ the world we live in, or from it, but ‘with’ it. Like color, perhaps.
Understanding the neural functions that contribute to the birth of mathematical structure and meaning is an active subject of research in cognitive science. A significant amount of work has been done to identify an innate ability we share with other creatures, namely the ability to perceive quantity. This is sometimes called our approximate number sense. Finding a path from this talent to how we come to use the symbolic representations of mathematics is certainly challenging. And a recent study published in Science contributes what is, perhaps, an unexpected observation. Researchers participating in the study observed that numerosity (defined as the set size of a group of items), while processed by the association cortex, mirrors the properties of primary sensory processing. Specifically, the neurons tuned to small numerosities follow topographical principles. These are principles associated with topographic representations that occur in vision, for example. In other words, neighboring points on the retina correspond to neighboring neural responses as the retina’s data is mapped to the cortex. The mapping is smooth, perhaps continuous in the mathematical sense. It is true that formal mathematical abilities, the ones we are trained to do, rely on different cognitive processes, but studies have found that individual differences in an innate number sense and other mathematical abilities are correlated. The Science paper not only suggests that topographic mappings may be active in higher-order cognitive processes, but also that number sense shares something with the primary senses.
It seems to me that the brain is a remarkable system that integrates everything that the body detects in sensation. And that the paths of this integration are continuous, from the most fundamental to the most complex. I recently had a conversation with my daughter about reading. For whatever reason, my daughter resists the idea that words on the page are symbolic representations of sounds. And this resistance inhibits her ability to read new words. There was a time when she needed to read out loud in order to fully comprehend what she was reading. And now, while she’s made significant progress reading to herself, it will happen that the words on the page aren’t making sense. But if she hears them, meaning is immediately comprehended. As we talked about it, I told her that her brain was doing an amazing thing when it translated something it was seeing into the meaning that was carried in sound. This action is another kind of integration of various cognitive processes. A systematic approach, contained in our biology, allows the body to build meaning over and over again. This conversation came to mind again today, when I read today about language in a recent piece on scientificamerican.com by Joshua Hartshorne. Hartshorne makes the following observation:
This design-constraint is interesting if you consider that there is likely a design-constraint on every aspect of cognition, on everything that we think we know. Perhaps this is a consequence of the way cognitive processes are consistently integrated. And, I would argue, that while mathematics is full of its own design constraints, the magic of what it does is found in the extent to which it can provide eccentric possibilities, or let more into the constrained system, by providing a uniquely thorough investigation of its own constraints.
The most recent issue of New Scientist has an article called Thoughts: The inside story. In it, philosopher Tim Bayne begins with a survey of all of the things we mean by the word ‘thought’ – the mental activity that accompanies perceptions, problem solving, the integration of various perceptions, the uncontrolled associative train of connected concepts, the organization of these associations, possibilities that are not perceived but fully imagined, and so on. Early on, Bayne makes the following remark:
I believe that this particular observation is worthy of more attention than it was given. And mathematics, given its role in the development of science, may be in a good position to contribute to the discussion. Mathematics is, in some sense, the conscious application of thought to perception. So how does the body make thoughts and what is it doing?
Bayne spends some time outlining why a “physicalist conception of thought” is more easily defended than an immaterial ‘mind’ or ‘soul’ conception, largely because of our observations of the significance of the brain’s role in thought. But, I believe even more to the point, he argues that “the materialist account of thought does justice to the continuity of nature,” and our relationship to other living creatures. The significance of the role that language plays in the presence of thought is somewhat undermined by studies that demonstrate thoughtful activity in non-human species – the chimpanzees’ ability to compare quantities and grasp simple fractions, or the baboon’s awareness of social hierarchies or the monkey’s ability to assess the difficulty of a task . While he didn’t mention it, the ability to discern quantity has been observed in fish as well. We are only beginning to grasp the complexity of non-human lives.
Language and symbol are here understood as facilitators of thought, perhaps providing for the unique sophistication of human thought, and the shared or social nature of our cognitive breakthroughs. Bayne quotes philosopher Andy Clark:
But Bayne seems to accept the usual bias about mathematical thoughts:
It is certainly true that mathematics is used to organize thoughts as well as perceptions, but mathematics doesn’t emerge as a ‘tool,’ it emerges as a ‘thought.’ And, I would argue that the birth of many mathematical thoughts is an action of the body, and not solely directed by our conscious will. Perhaps mathematics itself is an act of perception.
As the foundation of many of the sciences, it would seem to be in a unique position to reveal something about the relationship between our thoughts and the world in which we live.
Bayne also makes this observation:
The great pleasure that mathematicians feel about their discipline may be due to the fact that the source of mathematical thoughts, lies deep within the layers of our perceptive and cognitive processes, and their rise to consciousness is, in fact, not willed. Yet they invite us to make more of them, with very carefully directed reflection.
I appreciated Bayne’s consideration of the limits of thoughts.
The role that mathematics plays in science demonstrates the extent to which our images and ideas can exceed the limits of our perceptive abilities. While mathematics opens the door to, what still appear to be, limitless imagined structures, it also opens the door to the inaccessible reaches of our physical world. So what is thought, as an action of the body, designed to accomplish? And what might mathematics tell us about that?
One of the reasons that the nature of mathematics has been such an enigma, is that we associate it with thought, and we tend to distinguish thought from the physical world. We do find mathematics in natural structures – some of these beautifully represented in a film you may have seen called Nature by the Numbers. We’re somewhat familiar with the efficiency of honeycomb structures built by the honey bee. Their geometry maximizes robustness while minimizing weight. Honeybees execute the building of these honeycombs with great precision. Researchers have also seen mathematics in the behavior of insects. Ants calculate the path to food which takes the least amount of time rather than the one that is the shortest distance. Huffington Post reported on a study this past April.
In The Math Instinct, Keith Devlin made a nice survey of the mathematics in nature – the navigation feats of migratory birds, the logarithmic spiral of a falcon catching its prey and the mathematics of locomotion are some of his examples. Generally speaking, these observations highlight the pragmatic value of the non-symbolic mathematics. But pragmatic expectations might actually obscure the path to a new insight. Much of this blog is devoted to investigating the ubiquitous presence of mathematics, in living and non-living things, in order to raise some novel questions about the roots as well as the significance of mathematics in our human experience.
A recent story in a National Geographic blog describes the mathematical behavior of a fish, but the pragmatic motivation for the circular objects these fish create is not so obvious. Large (6.5-foot-wide) structures on the seafloor were once a mysterious feature of the underwater landscape. But these decorated circles have now been attributed to 5-inch long male pufferfish. The fish use their bodies to construct and decorate these nests, within which accepting females will lay their eggs.
The mail fish remains in the nest to supervise the eggs’ hatching. He then looks for a new site where he starts the nest making process again. It takes a significant amount of time for this small fish to build these large geometric structures, swimming toward the center of the circle in a straight line then around the center in a circular motion. But the pragmatic value of this geometry is not obvious. It reminded me of the bowerbird. David Rothenberg described the creativity of the bowerbird’s courtship structures in his book Survival of the Beautiful.
I enjoy both of them because they just don’t quite fit into the functionality-driven ideas that we use to describe our world. They seem to allow the possibility that life creates for no reason. Creations become tied to what look like reasons, but the creations are not fully captured by the reasons. This is the way I sometimes think of mathematics. One of the things that gets in the way of an informed appreciation of the human development of mathematics is pragmatism.
David Deutsch is proposing a very interesting conceptual shift in how we understand the nature of a physical theory. It’s an idea he has for “generalizing the quantum theory of computation to cover not just computation but all physical processes.” The theory in question he calls Constructor Theory. A video of Deutsch outlining the ideas (as well as the accompanying written text) were provided by Edge.org last October.
Constructor Theory rests on the nature of information is understood. According to Deutsch,
That information is physical, yet independent of the physical object in which it resides, is one of the cornerstones of the theory. Deutsch makes the point:
The theory works on yet another level of abstraction, (and a familiar mathematical idea) transformations.
The laws of Constructor Theory are expressed with an algebra.
Deutsch acknowledges that, while promising, the effort is new and if it turns out to be wrong, “Then we would have to learn the lesson of how it turned out to be wrong.” At the end of last year, Deutsch authored a paper that may be found here.
One of the things I most enjoy about David Deutsch is how he expresses his own captivation with how we have come to know things that lie far outside the range of our experience. In this particular talk he says the following:
I think the significance of this was failed to be appreciated by Arnold Trehub, a psychologist at the University of Massachusetts, Amherst and author of The Cognitive Brain. He contributed this critique of Deutsch’s ideas:
The provisional nature of knowledge is understood. But the constructive action of the brain is by no means being ignored in Deutsch’s ideas. The projection of that preconscious neuronal activity may be one of the things Deutsch’s method is meant to capture. That “volume of information signaling” in the brain, that is not a part of his conscious experience, may be the very thing for which mathematics compensates. Science (particularly the mathematical side of science) at the very least , suggests that the brain is terribly underestimated when it is characterized as “a pragmatic and opportunist organ.” The more interesting question would be how is mathematics and science contributing to the brain’s construction of its world.
As I read more discussions of the relationship between mathematics and physics, I find that what mathematics might reveal about how physical science progresses becomes an increasingly interesting question.
I recently found the text of a lecture given by Paul Dirac in 1939. It was reproduced on the occasion of the Dirac Centennial Celebration organized by the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 2002. The title of the lecture is “The Relation between Mathematics and Physics.” Dirac remarks, right away, that “there is no logical reason” why mathematical reasoning should succeed as one of the two methods used by the physicist to study natural phenomena (the other being experiment and observation). And, early in the talk, he says the following:
He goes on to survey the conceptual shifts that have happened in physics – from the equations that represent the laws of motion in Newtonian physics, to the geometry of Einstein’s space-time, and the non-commutative algebra of quantum mechanics. He made what I thought was an unexpected distinction between classical laws, governed by “a principle of simplicity,” and the mathematical beauty that makes the theory of relativity so compelling. About this Dirac says the following:
In this light he proposes that a powerful method of research for physicists may very well be to first choose a promising branch of mathematics “influenced very much in this choice by considerations of mathematical beauty,” and then proceed to develop it, keeping an eye on the way it lends itself to physical interpretation. String theorists are among those who seem to have chosen this route. And here’s an interesting statement:
‘The rules,’ of course, are the consequences of various observed relationships among the concepts captured in mathematical symbol. These may be relationships among numbers, or among operations and transformations, or among spatial properties and geometric structures. While there continues to be disagreement about whether the rules are invented or discovered , what Dirac may be emphasizing about mathematics is that the mathematician looks only at the mathematics, requiring no other validation of what mathematician Richard Courant once called ‘verifiable fact.’
Yet mathematics is an equal partner in the design of physical theories, what we consider a purely empirical science. This fact must say something about the nuances of what we mean by ‘empirical,’ which are often reflected in disputes between rationalists and empiricists. For some, the senses are like detectors to which a rational device of the mind is applied. But this perspective has been consistently challenged by research in cognitive science. These studies indicate strongly that the senses are not easily distinguished from that rational device. And perhaps it is this that is reflected in the growing blend of mathematics and physics.
Stephen Hawking was one of the speakers at the Dirac Centennial Celebration. The title given his talk was “Gödel and the end of physics.” But he wasn’t predicting the end of physics. The view he presents is, to some extent, a critique of the standard positivist approach to science, where the mathematical treatment of sensory data is the only source of knowledge. In this light, mathematics is not considered a product of the mind. Intuition and introspection, after all, play no role in the acquisition of knowledge. But how could one divorce intuition and introspection from mathematics? Physics’ increasing reliance on mathematics must be pointing to a relationship between mathematics and perception. And Hawking sees another problem. Physical theories, or mathematical models of physical systems, are self-referencing.
Like Gregory Chaitin, Hawking seems to find this incompleteness promising.
Kuhlmann is currently a philosophy professor at Bielefeld University in Germany and has dual degrees in physics and philosophy. I was happy to see that he is firmly committed to the idea that the task of understanding the physical world requires both disciplines.
Kuhlman takes the time to describe, in fairly simple terms, the content of the Standard Model which consists of groups of elementary particles and the forces that mediate their interaction. He describes how the particles blur into fields while, at the same time, the fields are quantized rather than continuous. His discussion of how the particles are not really particles and the fields are not really fields leads him to his point:
Kuhlman then takes his article in two interesting directions. The first is to focus on the notion of structure.
I was immediately reminded of a passage in the Courant/Robbins classic What is Mathematics? When I first read the book, I was impressed with implications of this observation which appears early in the text.
In the context of the Courant book, this is an important observation about the development of mathematics. But I have always thought that it can be seen as an important observation of a more general intellectual maturity. And this, I think, leads to Kuhlmann’s second alternative for interpreting the meaning of quantum physics which chooses ‘properties’ rather than ‘objects’ as having an existence.
This idea is consistent not only with current theories in cognition, but also has roots in 19th century philosophy and science (in the work of Hermann von Helmholtz and Johann Friedrich Herbart, for example). Kuhlmann rightly argues that our first experiences are of properties.
With respect to physics,Kuhlmann explains
The forgetting of ‘conceptual apparatus’ to which Kuhlmann refers is the very thing that I always hope (and expect) that mathematics will remind us of – in one way or another.