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By Joselle, on January 23rd, 2013 I’m not sure why I hadn’t been aware of it before today, but documentary film maker Ekaterina Eremenko has made the film Colors of Math. (Its Russian title is Sensual Mathematics)
I was happy to see that the work was supported and partly funded by my alma mater, The Courant Institute of Mathematical Sciences at New York University. Science Friday reported on the film in July, noting that it had not been taken up by an American distributor. Its US premier was at the Imagine Science Film Festival this past October.
The film’s synopsis is given in this way:
To most people math appears abstract, mysterious. Complicated. Inaccessible. But math is nothing but a different language to express the world. Math can be sensual. Math can be tasted, it smells, it creates sound and color. One can touch it – and be touched by it…
You can get a sense for the feel of the action and the character of the perspective from the trailer, which is almost six minutes long. These are some of the (perhaps paraphrased) observations that I particularly liked when I watched it:
…you put together all the ingredients and you mix and you mix and you wait for the chemical reaction….
At these sharp ends, the leaves don’t want to be flat
At every level even an abstract theory is some sort of space. Every notion and every action is about the interaction of geometric objects.
The film seems to want to capture something fundamental about mathematics. It is a view of mathematics that, despite all of our efforts, is still not very clear. And this fairly intimate look at it certainly contributes to the portrait of mathematics as natural, embedded in us, and in the world, more deeply than we have yet appreciated. Our senses, after all, build structure from the flood of interactions between our bodies and the world. And Colors of Math ties mathematics pretty tightly to the senses. Getting a feel for how this translation, from fleshy to symbolic, might be happening has been the primary focus of Mathematics Rising. Perhaps learning mathematics can be thought of as exercising the hidden organizing processes inherent in the body (or the brain). Some time back, these organizing actions somehow became part of our awareness, when we began to make symbolic images of our experience – in pictures, tallies, and clocks. Once given attention, they emerged further, themselves interacting, growing in substance, extending the senses. Maybe they are even turning the senses inside out.
By Joselle, on January 16th, 2013 Things happen in nature. Cells socialize and build structure, organisms grow, and move, and interact, and then more things grow – like music, language, and mathematics. Generally, talk about evolution is very pragmatic. Cell organization, the shaping of roots, leaves, nourishment mechanisms, reproductive drives, are all usually understood as fairly specific purposeful processes. Perhaps by sheer habit we extend this to our study of language and culture, even mathematics and music. But there is growing evidence that it would be worth our while to shake this habit. It may be causing us to overlook the significance of structure that continuously emerges from the very flow of life.
In a nice post at National Geographic Virginia Hughes discusses a recent study on the mechanisms that underlie music as was reported by the National Academy of Sciences. The study, by researchers at the University of Singapore’s Graduate Medical School, focuses on the relationship between music, movement, and emotion. It was motivated by psychology professor Thalia Wheatley’s earlier work, where she did neuroimaging studies which showed that brain regions involved in perceiving motion were some of the very same regions activated during music perception. This suggested that the two skills were linked. The University of Singapore researchers found a clever way to demonstrate that music and movement share a common structure that will produce equivalent expressions (both in animation and in sound).
The National Geographic post referenced an earlier piece written by Carl Zimmer that describes the work of Aniruddh Patel, an expert on music and the brain at the Neurosciences Institute in La Jolla, California. Patel investigated the relationship between music and communication.
It looks as if music is riding the coattails of other parts of the brain that evolved for other functions.
In a chapter of the recent book Emerging Disciplines, Patel describes how that borrowing process might work. Listening to the tones in instrumental music, for example, activates language regions of the brain that also process words and syntax. Those regions may make sense of tones by parsing melodies almost as if they were sentences. To keep a beat, Patel’s research suggests, we co-opt the brain network that links our hearing and the control of our muscles. This network’s main job is to allow us to learn new words.
Looking further I found a piece in The Guardian by mathematician Marcus du Sautoy about music and mathematics in which he makes the argument that what binds music and mathematics
is that composers and mathematicians are often drawn to the same structures for their compositions. Bach’s Goldberg Variations depend on games of symmetry to create the progression from theme to variation. Messiaen is drawn to prime numbers to create a sense of unease and timelessness in his famous Quartet for the End of Time. Schoenberg’s 12-tone system, which influenced so many of the major composers of the 20th century, including Webern, Berg and Stravinsky, is underpinned by mathematical structure. The organic sense of growth found in the Fibonacci sequence of numbers 1,2,3,5,8,13 . . . has been an appealing framework for many composers, from Bartók to Debussy.
And we all know that the Fibonacci sequence is expressed in a multitude of living forms like tree branches and shell spirals. Together these thoughts reinforced my sense that a pragmatic view of life is a weak and stingy view of things. I wrote about music, movement and mathematics in an early post. There I quoted from Fritjof Capra’s The Science of Leonardo.
Since Leonardo’s science was a science of qualities, or organic forms and their movements and transformations, the mathematical “necessity” he saw in nature was not one expressed in quantities and numerical relationships, but one of geometric shapes continually transforming themselves according to rigorous laws and principles. “Mathematical” for Leonardo referred above all to the logic, rigor, and coherence according to which nature has shaped, and is continually reshaping, her organic forms.
Mathematics is not just a way to describe the world around us. It must be, itself, another living expression of it.
By Joselle, on January 9th, 2013 Sean Carroll, Theoretical Physicist at the California Institute of Technology has recently published a new book. Entitled The Particle at the End of the Universe: How the Higgs Boson Leads us to the Edge of a New World it discusses the importance of the Higgs boson as well as the significance of the extraordinary work done at the Large Hadron Collider at CERN where evidence for the Higgs existence was finally found. The attention brought to Carroll from reviews and comments about the book, and the references to his exploration of time in physics, led me to look at some of his earlier work. I listened to a TED talk he gave last year on distant time. The talk was only about 15 minutes long, and one of the things that struck me was how concept-driven physic’s notion of time has become. It’s not exactly the experience of time that has physicists’ attention, but more whether (or how) it fits as a piece of the universe puzzle we’ve constructed.
Carroll’s description of time begins with the observation that entropy, the amount of disorder in a system, can be quantified. Entropy is described as “the number of ways we can rearrange the constituents of a system so that you don’t notice, so that macroscopically it looks the same. ” And so the reason entropy increases is simply because there are many more ways to be high entropy (more disordered) than low entropy (largely ordered). This is as much an appeal to numbers as it is to a physical system. With this, Carroll defines time:
Every difference that there is between the past and the future is because entropy is increasing — the fact that you can remember the past, but not the future. The fact that you are born, and then you live, and then you die, always in that order, that’s because entropy is increasing.
He goes on to describe how, in books and lectures, Richard Feynman emphasized that
The arrow of time cannot be completely understood until the mystery of the beginnings of the history of the universe are reduced still further from speculation to understanding.
Whether time is perceived or constructed is open to debate in physics today. I believe Carroll is of the opinion that it is perceived.
Inevitably I thought about how time has moved in and out of mathematics – how the derivative, for example can be thought of as a rate or as speed or movement along a curve, while at the same time mathematicians required that it be shown to have a purely arithmetic meaning. It is a conceptual challenge to us to say, outside of these many things it can be, what the derivative actually is. The structure that our biology gives to the world in experience inspires both concepts and processes in mathematics, but we consistently unravel mathematics from our experience in our search for indisputable meaning. The sciences, though defined by empiricism, necessarily follow paths that are opened with mathematics.
I also read an article today by Anil Ananthaswamy in New Scientist called Quantum Shadows The Mystery of Matter Deepens. I thought the article contained a related observation. In recent experiments that were designed to explore the wave/particle (dual) nature of light, an unexpected thing happened. The mixture of both aspects could somehow be seen.
… it took only a few months for the experimentalists to catch up with the theorists. But when three independent groups, led by Chuan-Feng Li at the University of Science and Technology of China in Hefei, Jeremy O’Brien at the University of Bristol, UK, and Sébastien Tanzilli at the University of Nice, France, performed different versions of the experiment last year, the results were unnerving – even to those who consider themselves inured to the weirdnesses of quantum physics (Nature Photonics, vol 6, p 600; Science, vol 338, p 634 and p 637).
…”Our experiment defies the conventional boundaries set by the complementarity principle,” says Li. Ionicioiu agrees. “Complementarity shows only the two ends, black and white, of a spectrum between particle and wave,” he says. “This experiment allows us to see the shades of grey in between.”
Near the conclusion of the article the author writes:
So, has Bohr been proved wrong too? Johannes Kofler of the Max Planck Institute of Quantum Optics in Garching, Germany, doesn’t think so. “I’m really very, very sure that he would be perfectly fine with all these experiments,” he says. The complementarity principle is at the heart of the “Copenhagen interpretation” of quantum mechanics, named after Bohr’s home city, which essentially argues that we see a conflict in such results only because our minds, attuned as they are to a macroscopic, classically functioning cosmos, are not equipped to deal with the quantum world. “The Copenhagen interpretation, from the very beginning, didn’t demand any ‘realistic’ world view of the quantum system,” says Kofler.
The outcomes of the latest experiments simply bear that out. “Particle” and “wave” are concepts we latch on to because they seem to correspond to guises of matter in our familiar, classical world. But attempting to describe true quantum reality with these or any other black-or-white concepts is an enterprise doomed to failure.
Paths like this one (where we can see something with certainty but have no way to say what it is we see) are remarkable. And it is mathematics that takes reveals them. That we have not yet, in any significant way, looked back at ourselves and wondered at what we are doing is equally remarkable. I’ll close with this from the article:
It’s a notion that takes us straight back into Plato’s cave, says Ionicioiu. In the ancient Greek philosopher’s allegory, prisoners shackled in a cave see only shadows of objects cast onto a cave wall, never the object itself. A cylinder, for example, might be seen as a rectangle or a circle, or anything in between. Something similar is happening with the basic building blocks of reality. “Sometimes the photon looks like a wave, sometimes like a particle, or like anything in between,” says Ionicioiu. In reality, though, it is none of these things. What it is, though, we do not have the words or the concepts to express.
By Joselle, on December 31st, 2012 I have always been intrigued by the extraordinary insights of the self-taught mathematician Srinivasa Ramanujan. He worked in almost complete isolation from the mathematical community, and independently rediscovered many existing results while also making his own unique contributions. He didn’t even share notation with the rest of the community, somehow finding his way without being led. I’m convinced that this remarkable life must be showing us something about the very nature of the thoughts he followed – something we have neglected about the nature of mathematics itself.
He was brought to my attention again when Scientific American wrote about India’s response to his 125th birthday on December 22. Last year Prime Minister Manmohan Singh declared 2012 to be a National Mathematics Year in India in honor of Ramanujan.
But then, even more newsworthy, I found a number of reports about how mathematicians were able to show that a hunch Ramanujan had about the properties of a class of functions (that were never before heard of) was correct. The story was reported in the Daily Mail on December 28.
While on his death-bed in 1920, Ramanujan wrote a letter to his mentor, English mathematician G. H. Hardy, outlining several new mathematical functions never before heard of, along with a hunch about how they worked.
Decades years later, researchers say they’ve proved he was right – and that the formula could explain the behaviour of black holes.
‘We’ve solved the problems from his last mysterious letters,’ Emory University mathematician Ken Ono said.
In each of the accounts of this development, some reference was made to the fact that Ramanujan’s insight was contained in a dream. From Daily News and Analysis:
Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri. However, no one at the time understood what he was talking about.
The same statement appeared in the Daily Mail with an image of the goddess.
A more thorough discussion of Ramanujan’s insight can be found in the article What is a Mock Modular Form? published by the American Mathematical Society.
From the Huffington Post:
Ramanujan believed that 17 new functions he discovered were “mock modular forms” that looked like theta functions when written out as an infinte sum (their coefficients get large in the same way), but weren’t super-symmetric. Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.
Ramanujan died before he could prove his hunch. But more than 90 years later, Ono and his team proved that these functions indeed mimicked modular forms, but don’t share their defining characteristics, such as super-symmetry.
The expansion of mock modular forms helps physicists compute the entropy, or level of disorder, of black holes.
In developing mock modular forms, Ramanujan was decades ahead of his time, Ono said; mathematicians only figured out which branch of math these equations belonged to in 2002.
“Ramanujan’s legacy, it turns out, is much more important than anything anyone would have guessed when Ramanujan died,” Ono said.
I enjoyed this video posted on youtube that helps bring both the math story and the personal story to life. It was suggested here that, given Ramanujan’s religious life, it isn’t really a surprise that he would attribute his vision to a Hindu goddess. But a statement like that is just suggesting that we needn’t think about it. There is something to think about, and it’s not whether mathematics is really divine or not. While there may be no easy way to address this question, the peculiarities of Ramanujan’s work should encourage some of us to wonder about how spiritual vision, dream vision, waking vision, and what I’m tempted to call cognitive vision (the perception of pure structural meaning) are related.
By Joselle, on December 17th, 2012 One of the points I wanted to make in last week’s post was that studies in animal cognition suggest the presence of mathematics in the behavior of non-human species – the ants, for example, who can be seen to pass on quantitative information to other ants. We don’t see the mathematics they may be doing because the form it takes is internal to them and their community. A paper that I referenced, by Zhanna Reznikova and Boris Ryabko (Numerical competence in animals, with an insight from ants) describes studies where it was possible to observe ants communicating the location of a food source using what appeared to be arithmetic.
Informally, the ants were forced to develop a new code based on simple arithmetic operations, that is, to perform an operation similar to passing the ‘name’ of the ‘special’ branch nearest to the branch with the trough, followed by the number which had to be added or subtracted in order to find the branch with the trough.
(This paper can be found on Reznikova’s website under publications)
This past Thursday, Diane Rehm brought my attention to the work of biologist Con Slobodchikoff, author of a new book, Chasing Dr. Dolittle. A transcript of the interview can be found here.
On his own website his prairie dog studies are described in this way:
The prairie dog work suggests that prairie dogs have a complex communication system that borders on language. They have different alarm calls for humans, coyotes, domestic dogs, and red-tailed hawks. In addition, the prairie dogs can describe the size and shape of an individual predator. This is the most sophisticated animal language system that has been described to date.
One of the points that Slobodchikoff was making in the interview was that one of the reasons we think that animals don’t have language is that we have not had a way to witness it, no less investigate it. But Slobodchikoff found a way with prairies dogs and in the interview he made an important distinction between communication and language.
Well, here’s the distinction actually. Simple communication is just the production and reception of signals. So a radio does simple communication. There’s no thinking involved. Language, on the other hand, has many different components. One of the components is it has flexibility. So the animal can decide what kind of signal to send in a particular situation. It’s not a fixed sort of thing. It’s not you push this button you get this output. You push that button you get a different output. The animal can decide.
The animal also can intend to send the signal. So there’s intentionality. There’s also novelty. The animal can make up new signals or combine signals in different ways that we haven’t seen before. So there are these kinds of different elements. There’s also structure in the sense of there’s usually a grammar associated with that. We have these kinds of elements that go into language that we don’t have in a simple communication system. So in a sense language is a subset of communication. But most scientists have not been willing to give animals the benefit of the doubt and say that animals have language. They are only willing to say that animals can communicate.
But not Slobodchikoff.
And I would say that many animals within that larger frame of communication actually have language using those features that I mentioned. And so if we are able to decode that language then we might, perhaps, be able to communicate with them on a level that they can understand.
What first struck me about the discussion was that Slobodchikoff was suggesting that linguistic structure could exist within an animal’s physiology and behavior (like the head bobbing and back-flattening of a lizard) which would look nothing like our own vocal and written symbols but could involve complex thought. Some of the strength of his conviction rests on what has been termed evolutionary continuity.
But if you think about the way that we are related to, say, other vertebrates, the fish, the mammals, the birds, the reptiles and so on. We all have a spine. We all have a circulatory system. We all have a nervous system. We all have a brain. And so if you extend all of that then you realize that language production or sound production or visual production is all part of a system where, for example, you have a signal coming into receptors.
The receptors send this along the nervous system to the brain. The brain makes a decision about what to do. And then sends signals along the nervous system to sound production organs or to visual production organs and so on. And we have that. And other animals have that, too. And so what I’m arguing with the discord system theory is that there is evolutionary continuity in all of these morphological structures and all of these physiological structures that we have and that other animals have, as well.
In a very positive review of the book, on the Psychology Today website, evolutionary biologist Marc Bekoff makes these references to the text:
Chapter 2 of “Chasing Doctor Dolittle” called “What is Language” lays out the core of Slobodchikoff’s argument. In a nutshell, he uses linguist Charles Hockett’s thirteen design features of human language (pp. 20 ff) and shows how nonhumans share them with us. He concludes this chapter by writing “I show that we already have the evidence to conclude that a number of animal species have semantic signals and that these signals are arranged according to rules of syntax within different contexts.” (p. 35) He then goes on to provide numerous examples of animal language.
I am intrigued by the idea that we might soon see that language is not linked to the tongue as we may have thought, nor to our symbolic expression of words. Rather it is the work of the body that the symbol accomplishes. And these blogs travel a similar road with respect to mathematics. Work like Slobodchikoff’s has the potential to change what we understand language to be, as would similar insights change how we see the fundamentals of mathematics.
By Joselle, on December 10th, 2012 I have recently spent some time sorting out the points Arkady Plotnitsky makes about the significance of Riemann’s notion of manifold (or manifoldness) in his paper which appeared in the journal Configurations in 2009. The paper has the title Bernhard Riemann’s Conceptual Mathematics, and the Idea of Space. It is refreshing in that it considers relationships among mathematical, philosophical, and ordinary thought. Plotnitsky explores the idea that there is no single form of thought, and suggests that Riemann’s work is a powerful example of the practice of this plurality in mathematics. This aspect of the paper came to mind today when I looked at two recent publications concerning non-human animal cognition.
The first of these is called Quantification abilities in angelfish: the influence of continuous variables by Luis M. Gomez-Laplaza and Robert Gerlai.
Previous studies with angelfish have demonstrated their capacity to discriminate quantities similar to the abilities found in other vertebrates. But this paper reports on studies that were done to correct the failure of earlier studies to control for the non-numerical features of the schools of fish (referred to as shoals) to which these angelfish respond (such as the density of the school, the interfish distance, and the overall space the shoal may occupy). In their concluding discussion they continue to qualify the studies’ findings:
As previously alluded to, the present results on the use of continuous variables for the discrimination of large and small quantities should be considered with caution for two main reasons. One, there are many visual features of a shoal that fish could potentially use to discriminate between shoals without having to rely on numerical information, and not all of these features were controlled in the current study. For example, fish could perceive such quantitative properties as surface area, contour length, amount (intensity) of light reflected, or total amount of movement in the shoals rather than relying on a strict form of numerical competence… it is possible that unlike experimenters, fish may not make dichotomous decisions and may use different types of information in a graded manner. That is, depending on the size of the shoals contrasted and depending on the availability of the different numerical and non-numerical types of information, the experimental subject may use one or the other feature(s) with differing weights. Clearly, understanding this complexity will require further research in which most, if not all, possible factors are systematically varied one by one and also in combination with each other. (emphasis my own)
With my mood set by reading about Riemann, those quantitative properties – like surface area, contour length, intensity of reflected light, and total amount of movement – called to mind Riemann’s careful thoughts about magnitude, and many of Plotnitsky’s points about the conceptual nature of Riemann’s mathematics. These properties are things perceived. I’m working on my own paper to suggest that Riemann’s insights could be viewed as insights into cognition itself.
The fish paper’s references led me to a paper by Zhanna Reznikova and Boris Ryabko published last year called Numerical competence in animals, with an insight from ants which can be found on Reznikova’s website under publications. Here, again, the emphasis is on language independent mental magnitudes. The paper surveys studies done on monkeys, pigeons, parrots, crows, dolphins, bees, beetles, fish, dogs, and human infants.
Advanced numerical abilities—exact counting and arithmetic operations— have been considered for a long time to be uniquely human and based on our language symbolic representation. Recent studies provide intermediary links between numerical abilities in human and non-human animals.
But the authors of this paper draw attention to a distinguishing feature in these studies: “all known experimental paradigms for studying numerical competence in animals do not exploit the phenomenon of close relations between intelligence, sociality and natural communication.” They look then to studies with ants. Ants were seen to communicate some kind of numerical information about the location of food.
In the described experiments scouting ants actively manipulated with quantities, as they had to transfer to foragers in a laboratory nest the information about which branch of a ‘counting maze’ they had to go to in order to obtain syrup…
The likely explanation of the results concerning ants’ ability to find the ‘right’ branch is that they can evaluate the number of the branch in the sequence of branches in the maze and transmit this information to each other. Presumably, a scout could pass messages not about the number of the branch but about the distance to it or about the number of steps and so on. What is important is that even if ants operate with distance or with the number of steps, this shows that they are able to use quantitative values and pass on exact information about them.
I’m struck by the potential here. This emphasis, in animal cognition studies, to look within the animal’s physical and social structures for a different kind of quantitative awareness, could contribute to how we understand our own sense of magnitude and quantity, which is now objectified in mathematics.
By Joselle, on December 3rd, 2012 I listened last week to Diane Rehm’s interview with Ray Kurzweil, author of the book “How to Create a Mind: The Secret of Human Thought Revealed” A transcript of the interview can be found here.
Published in mid-November, it is already a New York Times bestseller, and some of the responses to it from prominent colleagues introduce the book’s website. While it is an exploration of the potential for technological advancement in the computer driven field of artificial intelligence, it does this in the light of new opportunities to understand how we might reverse-engineer the brain. It was these insights into how the brain learns or how it builds our symbolic worlds, that caught by attention, looking to me again very much like mathematics.
During the interview, Kurzweil talks about the neocortex which, while present in mammals, is uniquely developed in humans.
We can take a whole bunch of ideas, call that a new idea, give that a symbol and then use that symbol with yet other ideas and create another symbol and build up this whole hierarchy we call knowledge. Only the neocortex is able to do that. And it enables us to solve problems and learn new skills that are flexible.
Our neocortex can create metaphors so Darwin noticed a metaphor from geology that tiny, little trickles of water can carve out a great cavern.
The efficacy of symbol in the development of thought is the very life of mathematics, and metaphors are how we move around within these structures.
Kurzweil argued in a paper that he wrote at the age of 14 (!) that the heart of human intelligence is recognizing patterns. This is what the brain does well.
that is actually the message of this new book 50 years later. But I describe in real detail how the neocortex does that. We have basically 300 million little regions that recognize patterns and they’re organizing hierarchies. So I’ll have one little pattern recognizer that fires when it–when it sees a crossbar in a capital A. And that’s all it cares about. It gets very excited when it sees a crossbar in a capital A.
At a higher level there’ll be a recognizer that recognizes a capital A. At a yet higher level there’ll be a recognizer that goes ah, it’s the word apple. Go up another 20 levels there’ll be a recognizer that’s getting input from different senses. It sees a certain fabric, smells a certain perfume, hears a certain voice and goes ah-ha, my wife has entered the room. Go up another 20, 30 levels as a recognizer goes, that’s funny, she’s pretty. That was ironic.
You might think that those higher level more abstract recognizers are more complicated, more sophisticated. They’re actually basically the same except for their position in this grand hierarchy. And that’s the essence of the neocortex. It’s organized in this very elaborate hierarchy from very low level primitive — recognizes — it just recognizes the edges of objects and simple things up to these very abstract features.
I’m intrigued by the idea that these recognizers don’t differ in any way other than position. They are essentially content-less, like mathematical symbols, and acquire meaning only in relation to each other. He goes on to say that the key is that we’re not born with this hierarchy:
We create that from our own experience. Everybody’s hierarchy is different because you are not only what you eat, but you are what you think. With every thought we make we create that elaborate hierarchy…But literally from the moment you’re born or even earlier — because our eyes open at age 26 weeks — we are laying down one conceptual level at a time. That’s another key is that we can only sort of learn one level of abstraction at a time. So I’ve got a one-year-old grandson now and so he’s managed to lay down a few levels. But he’s, you know, still learning the primitive features and patterns and language and so on.
And why is it that we might encounter so much difficulty when we try to see the more elaborate hierarchies of advanced mathematical ideas? Perhaps it’s because we’re not motivated to make room for them.
…we actually fill up these 300 million pattern recognizers by the time we’re 20. And in order to learn something new at that point we need to forget something old…Now some people are better at giving up old ideas than others. I mean, many people are rigid thinkers and just happy with the neocortex they have and don’t want to entertain new ideas…It’s comforting to have explanations for everything. We have to realize the limitations of our ideas to gain new ones.
I may be in the minority, but I really like discovering the limitations of our ideas. In the interview, Kurzweil also made a good argument for why we may be built this way, why the neocortex grew this kind of functioning – because our thoughts are like everything around us.
The world is hierarchical and you really can’t understand the world unless you can understand its natural hierarchy. The fact that, you know, branches lead to other branches and the whole assembly of branches makes a tree and this assembly of trees make a forest. And this is a natural hierarchy to the world. You can’t really understand it otherwise. This was the first mental organ that enabled animals to do that.
I have often suggested that mathematics happens, that we don’t actually descide to make it. Yet intentionality plays a powerful role. Kurzweil addressed the enigma of consciousness as well.
Most of what goes on in the neo-cortex and that is where we do our thinking is not conscious. We’re only consciously aware of a small part of it and that actually gets at the issue of responsibility and free will…But I cite some interesting studies where people actually begin to carry out a decision before they’re even consciously aware that there was a decision to be made. And this was very, very interesting research. It’s well-designed and people became aware that there was a decision after they already carried out the decision. So this brings up the mystery of consciousness.
I have long thought that mathematics may be bringing the less conscious aspect of thought to our awareness.
There are numerous press responses to this latest from Kurzweil that can be found on the book’s website.
By Joselle, on November 26th, 2012 Having heard the clip from Spielberg’s latest film, Lincoln, where Lincoln describes Euclid’s first common notion, I tried to investigate the extent to which the connection between Lincoln and mathematics has been pursued, and I was disappointed. It’s difficult for anyone to speak about mathematics without sifting out the structure, reason and proof that characterizes mathematics, from its life-sustaining meaning. Life-giving meaning is reserved for things like poetry, music, and art. Every attempt to consider Lincoln’s fascination with Euclid falls into this trap, despite the fact that Lincoln, himself, seems to have avoided it. Euclid’s first common notion, as described by Lincoln in the film, says that things that are equal to the same thing are equal to each other. The purpose of the reference must be to get us to wonder about how this rational statement about equality would be relevant to the slavery issue that defined the crisis of Lincoln’s presidency.
Lincoln’s preoccupation with Euclid inspired the book Abraham Lincoln and the Structure of Reason by David Hirsch and Dan Van Haften. The authors describe how their work took shape in an interview on the book’s website, where the authors claim to have cracked the code of Lincoln’s speeches, identifying the underlying structure that made them beautiful and effective. But a review of the book, by author and historian Jason Emerson, points to the weakness of a purely analytic approach to how mathematics influenced Lincoln.
…it is derelict of the authors to include no consideration of how Lincoln’s logical and mathematical mind could have impacted his presidency, such as how the logic in his speeches allowed him to convince the public of the correctness of his policies and to therefore lead them in the direction he wanted them to go; his rationalizations for legal issues such as suspension of habeas corpus or the issuance of the Emancipation Proclamation; his decisions on military tactics and planning; his penchant for technology and innovation in weapons, medicine, transportation, and communication, which is all a direct corollary to his logical and technically-inclined intellect.
Perhaps the nature of Lincoln’s own inquiry quoted on the website Math Open Reference, points to the enigmatic connection:
“In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?
I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.
At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.”
So, what did he find out? Mathematics revealed a kind of demonstration the meaning of which was not fully captured by all the dictionaries and books of reference Lincoln consulted. It seems clear that Lincoln was not interested in making an analogy between mathematical truth and living truth. He followed a path carved out by mathematics to clarify his own vision, and then he communicated what he saw using some of the same structure in the speeches he gave. He chose an arrangement of words, words that were tied to the experience and hopes of the people around him, and moved many of them to see what he saw. I would suggest that in order to do this, he was guided, not only by the structure of Euclid’s reasoning, but also by its content. And this is where our difficulty in understanding the development of Lincoln’s thoughts begins. The difficulty we have in fully understanding the impact of Euclid on Lincoln highlights the difficulty we have understanding the nature and power of the insight mathematics can provide. Mathematics, as most people see it, looks like structure more than content. But I consistently argue that mathematics produces insight because of the way it is tied to the cognitive processes that, from the bottom up, build our lives.
Doris Kearns Goodwin, the author and historian on whose book the film Lincoln is partly based, took note of some of Lincoln’s passions in an interview with Charlie Rose. She tells us that Lincoln, a mostly self-taught man, was excited to the point of not sleeping, to read Shakespeare and the King James Bible. I suspect that Lincoln read Euclid the way he read everything else, to see more of what could be seen. It was at once personal and a path to truth. And it is this that has been lost in our modern approach to mathematics education, understanding, and appreciation.
By Joselle, on November 12th, 2012 Our thoughtful, imaginative worlds are married to our physical experiences but the subtleties of their union are almost impossible to fully appreciate. Mathematics, I often argue, has the potential to provide a better view of the situation, perhaps because of the inexhaustible depth of its abstraction, together with the precision it brings to a concept, and some of the surprising fruits of its application. A recent paper from Roy Wagner at The Hebrew University of Jerusalem comes at it, again, from a more formal, philosophical perspective rather than a cognitive science perspective. The paper, Infinity Metaphors, Idealism, and the Applicability of Mathematics, highlights the plurality of conceptual frameworks within mathematics, particularly with respect to infinities, and uses this to make an interesting point. Wagner quickly addresses the book Where Mathematics Comes From from George Lakoff and Rafael Nunez and surveys the short-comings in how they account for the notion infinity. He produces mathematical notions of infinity that can’t be captured by the Basic Metaphor of Infinity (BMI), proposed by Lakoff and Nunez in their naturalistic account of mathematics. This BMI is the cognitive mechanism underpinning our concept of infinity in mathematics. The idea is that the concept of continuous processes is a broadening of our experience of iterative processes, processes that go on and on. Among other examples, Wagner points out that the uncountable infinity of real numbers is not captured by the BMI.
Uncountable, strongly inaccessible cardinals are defined in such a way that they cannot be constructed from smaller cardinals, namely, they cannot be the end state of any infinite process constructable from Aleph0. In a way, they are conceived, precisely, as that which cannot be conceived through BMI! The metaphor here is not that of a final state of an indefinite process, but of that which is beyond any final state of any indefinite process, including the indefinite process of constructing ever larger infinities – a figure of transcendence that’s not captured by BMI.
Wagner also points out that, while Lakoff and Nunez apply this iterative process idea to the point at infinity in projective geometry, it doesn’t actually work.
To make one final point, let’s consider one more example. Henderson (2002) observes that explaining the projective point at infinity in terms of BMI (as Lakoff and Núñez do) would yield two intersection points at infinity for any pair of parallel lines (one point in each direction). So BMI cannot be the origin of the single projective point at infinity. Indeed, the origin of this point at infinity is easy to track down: it comes from the point of perspective in Renaissance art and draftsmanship – a practical tool, rather than an inference carried between domains. While expressing a commitment to the notion of embodied cognition, Lakoff and Núñez’ notion of metaphor largely ignores the origin of mathematical ideas in culturally constrained practices with tools.”
(The Henderson reference is to a review of the Lakoff/Nunez book that appeared in the Mathematical Intelligencer in 2002).
I was happy to see Wagner point to the investigation of perspective in Renaissance art as the origin of the point at infinity. In this light, the idea looks like a tool rather than a metaphor, and this will support the thesis Wagner later argues. But even this doesn’t do justice to what captivates me about this development in mathematics. The point at infinity completes the mathematical idea of parallel vs. not parallel by examining an infinity that can’t actually be reached. (which had been one of the difficulties with Euclid’s parallel postulate). At the same time, the source of inspiration for this examination came from th visual experience of that same receding distance, from how it appears to the eye.
A large part of Wagner’s paper is a discussion of the work of 19th century Polish philosopher and mathematician Jozef Maria Hoene Wronski. When Wagner begins to sum things up he says this:
Wronski’s form of idealism (like other forms of German idealism, but with a mathematical edge) thinks of being not as given, but as formed by the knower, who is in turn formed by that being. Of course, neither being nor knowledge are formed arbitrarily, but being is carved out with the tools and concepts of the knower…Tools and concepts enable humans to inhabit the world in different ways and carve out different observations and forms of understanding. It is in this sense that reason can constitute and impose on being, rather than content itself with a regulative role. As knowledge evolves, so do our tools, ways of intuiting, and the phenomena we encounter and create.
While this is a philosophical piece, it certainly brings to mind the perspective in biology that was pioneered by Humberto Maturana and Francisco Varela. In their book, The Tree of Knowledge. http://www.shambhala.com/the-tree-of-knowledge.html they describe cognition as “an ongoing bringing forth of a world through the process of living itself.” But in Wagner’s paper this idea has an interesting consequence.
This approach allows us to extend the plurality inherent in mathematical language to what can be conceived as a plurality inherent in the ways humans form being. Instead of a single “mathematics” miraculously corresponding to a single “physics,” we have an inherently multiple mathematical language that corresponds to some way of inhabiting our universe.
This suggests that there are ways of inhabiting our universe that we do not yet see, but may already have been given shape in mathematics, universe in our imagination.
Describing mathematical knowledge as part of the physicists’ instruments span, Wagner characterizes physics in a way that Galileo could not have forseen. He proposes that
Physical phenomena are not simply observed by physicists, but also constituted by the possibilities that the physicists’ instruments span…In a way, modern physics is inherently designed to discover precisely that portion of our way of inhabiting the universe that can be discovered through mathematical analogies.
By Joselle, on November 5th, 2012 I found a short paper today by Mark Andrews, Stefan Frank and Gabriella Vigliocco focused on reconciling two trends in the study of meaning in cognitive science. These two trends are represented by embodied cognition theories (which treat meaning as a simulation of perceptual and motor states) and by computational or distributional accounts of meaning (where meaning is seen as a consequence of the statistical distribution of words across spoken and written language). The authors explain:
Clark (2006) has presented a compelling account of how language can be seen as another (literally) physical environment that agents may perceive and act upon. By this account, amongst other things, language provides a new source of perceptual data and new targets for action, in a manner identical to that of any other modality. It provides “a new realm of perceptible objects. . .upon which to target (our) more basic capacities of statistical and associative learning” . In other words, language is an extension of the physical environment generally, and one that we may perceive (by language comprehension) and act upon (by language production), just as we do with any physical environment.
There is also growing evidence in support of Clark’s (2006) hypothesis that we use language as a model of, or proxy for, the world. In fact, this is highly related to the symbol interdependencyhypothesis of Louwerse (2007, 2011; Louwerse & Jeuniaux, 2008) that describes language as encoding relations in the world, including embodied relationships. By this account, we can learn and represent the world by way of learning these intra-linguistic relationships. Louwerse (2011) describes how the perceptual relationships underlying the modality-switching task of Pecher et al. (2003) (see Section 2.1) could be learned from distributional relationships in language. Louwerse (2008) similarly shows that spatial relations are encoded in word-order statistics.
The paper argues that sensory-motor data and intra-linguistic distributional data are interdependent – that something like sensory motor action is happening within the symbolic construction of language. And I think that this debate in linguistics can also serve to highlight some of the questions cognitive scientists and philosophers have had about what mathematics is and how it manages to do what it does. Mathematics has transcended what may be seen as its concrete roots (quantity and space) and looks like it has become completely dis-embodied. But it has grown into one of humanity’s most productive, yet purely abstract systems of thought.
Guy Dove, a philosopher at the University of Louisville wrote a paper last year for Frontiers in Cognition that presented an argument echoed by the Andrews paper. He says early on:
This essay proposes and defends a pluralistic theory of conceptual embodiment. Our concepts are represented in at least two ways: (i) through sensorimotor simulations of our interactions with objects and events and (ii) through sensorimotor simulations of natural language processing. Linguistic representations are “dis-embodied” in the sense that they are dynamic and multimodal but, in contrast to other forms of embodied cognition, do not inherit semantic content from this embodiment. The capacity to store information in the associations and inferential relationships among linguistic representations extends our cognitive reach and provides an explanation of our ability to abstract and generalize.
This last observation, that the capacity to store information in associations and inferential relationships extends our cognitive reach, is particularly true of mathematics.
Dove points out that the orthodox approach to concepts posits that there is no intrinsic connection between symbols and what they represent. But this presents the problem of their grounding, of symbols related to each other but to nothing else.
Perhaps the easiest way to think of this problem is to imagine trying to learn a foreign language from a dictionary in that language. Each word would be defined in terms of its connections to other words.
A well-known limitation of the evidence for embodied concepts is that it primarily involves concrete or highly imageable concepts (Pezzulo and Castelfranchi, 2007; Louwerse and Jeuniaux, 2008; Dove, 2009). This is problematic because, although it is not difficult to imagine how embodiment might help us acquire concrete concepts, it is difficult to see how it can be anything but a hindrance with abstract concepts such as DEMOCRACY, ELECTRON, ENTROPY, JUSTICE, NUMBER, PATIENCE, and TRUTH. Representations grounded in sensorimotor systems do not seem to be well suited to representing abstract intentional contents. For this reason, abstract concepts remain a critical issue for embodied cognition. More is at stake than simply the reach of this approach. For instance, Mahon and Caramazza (2008, p. 60) use the challenge posed by abstract concepts to support a parsimony argument in support of an amodal approach to concepts:
Given that an embodied theory of cognition would have to admit ‘disembodied’ cognitive processes in order to account for the representation of abstract concepts, why have a special theory just for concepts of concrete objects and actions?
Dove surveys how various embodiment theories of language account for abstract concepts. Included in his review are the conceptual blends, accomplished through metaphor, that are explored in the Lakoff/Nunez book Where Mathematics Comes From. He finds them all inadequate and proposes an alternative.
I suggest that language plays two roles in our cognitive lives. One role is to engage sensorimotor simulations of interacting with the world. In this role, language serves primarily as a medium of communication. A second role is to elicit and engage symbolically mediated associations and inferences. Our concepts are not merely couched in sensorimotor representations but also in linguistic representations (words, phrases, sentences). Conceptual content is captured in part by the relationships of linguistic representations with other linguistic representations. These relationships may be merely associative or they may be inferential…This philosophical theory of mental content holds that the meaning of a concept is determined by its functional role within the cognitive life of an individual.
One of the ways Dove defends this possibility is with the effectiveness of distribution models where the meaning of words can be derived through statistical computations applied to large bodies of text.
A couple of previous blogs come to mind here. One from last year where. within a discussion of Leibniz, Herbart, and Riemann, I described Herbart’s eccentric belief that idea’s themselves (in this context represented, perhaps, by abstract concepts) struggle to gain expression in consciousness – that they actually compete with each other to do so. This might sound odd to the ears of modern science, but what really distinguishes this notion from ours is that it gives thoughts life. This is in contrast to seeing thoughts as the output of a system or process. And this brings another blog to mind, one that highlighted the work of physicist Bob Coecke. Coecke made the observation that the way words interact in a sentence (to create meaning) is similar to the interactions in the subatomic world of quantum mechanics. Coecke has developed a mathematics that can simplify calculations involving quantum mechanical processes and he has had some success applying these methods to the study of language.
This whole blog can look like a dish of spaghetti, a little mess of overlapping strings of thoughts. But all of these discussions of meaning and cognition dig deeper into how we see in our symbolic worlds. And mathematics is exactly a thorough exploration of symbolic worlds. What seems evident in these studies is that the weave of perceptual meaning and conceptual meaning is not easily disentangled. Their observations help blur the distinction between the physical and the mental. And it is on this path that I think mathematics can do a lot to light the way.
Just for fun, I want to add that when I started writing this morning, I overheard Alex Danchev, author of the new book Cezanne: A Life, during an NPR interview. He said that Cezanne was looking for the truth, trying reconstruct or re-present the world with not only what he saw, but also what he felt.
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