My post appeared on the Scientific American Guest Blog this morning. Here’s the link:
My post appeared on the Scientific American Guest Blog this morning. Here’s the link:
My time this week is again taken up with work on a few writing projects that I’m trying to wrap up (not to mention end of the term grading). But I should be back on track with my regular blogs next week.
In the meantime, an article on scientificamerican.com caught my attention, being about the mathematics of juggling! But the article originated elsewhere, at Simons Science News. And this is how I became aware of the Simons Foundation website. It’s worth exploring.
It is a private foundation that provides some interesting grant opportunities.
The site provides some video interviews with mathematicians organized under the heading Science Lives. These interviews include ones with John Nash, Michael Atiyah and Cathleen Morawetz from my own alma mater – The Courant Institute of Mathematical Sciences. There’s also an interesting article on new observations in biology that are consistent with a mathematical idea of Turing’s proposed in 1952.
I’m short on time today and working on a guest blog which I hope to be able to provide a link to shortly. But I did begin exploring a website that has short video interviews with some of my favorite thinkers. I found among a list of participants on the website Closer To Truth, Gregory Chaitin and David Deutsch. They have each participated in a number of video interviews that seem always short, but always interesting.
I listened to one of Deutsch’s this evening called What is Ultimate Reality? In the interview he described the four fundamental, interdependent aspects of reality, as presented in his book, The Fabric of Reality. These are, quantum physics, the theory of evolution, the theory of computation and the theory of knowledge. Briefly, he says that quantum physics constrains the kinds of theories that one can express and that evolution is the theory of emergent properties that cannot be expressed in terms of atoms. Computation is about the processes in nature that are independent or that transcend the substance in which they are embodied. And knowledge is the kind of information that can do things. Knowledge is embodied in DNA, brains, books, computers, etc. But Deutsch makes the point that what moves things, what changes things or creates things is information, not the substances in which the information is embodied.
For me, the most refreshing and important thought was his observation that all of these aspects of reality have been underestimated by being accepted as the right explanation in their own field. But they have not become integrated. The depth of these specialized fields has made it increasingly more difficult to consider their interdependence.
In another interview, Gregory Chaitin explores his own Platonism.
I recommend listening.
I was expecting to write about a paper I found recently by Oran Magal, a post doc at McGill University, On the mathematical nature of logic. I was attracted to the paper because the title was followed by the phrase Featuring P. Bernays and K. Gödel
I’m often intrigued by disputes over whether mathematics can be reduced to logic or whether logic is, in fact, mathematics, because these disputes often remind me of questions addressed by cognitive science, questions related to how the mind uses abstraction to build meaning. This particular paper acknowledges, in the end, that its purpose is two-fold. It makes the philosophical argument that an examination of the interrelationship between mathematics and logic shows that “a central characteristic of each has an essential role within the other” But the paper is also a historical reconstruction and analysis of the ideas presented by Bernays, Hilbert and Gödel (the detail of which is not particularly relevant to my concerns). It was Bernays’ perspective that I was most interested in pursuing.
Magal begins with the observation that
While some have seen logic as more general than mathematics, there has also been the view that mathematics is more general than logic. It is here that Magal introduces Bernays’ idea that logic and mathematics are equally abstract but in different directions. And so they cannot be derived one from the other but must be developed side-by-side. When logic is stripped of content it becomes the study of inference, of things like negation and implication. But while logical abstraction leaves the logical terms constant, according to Bernays, mathematical abstraction leaves structural properties constant. These structural properties do seem to be the content of mathematics, and what makes mathematics so powerful.
Magal describes how Bernays understands Hilbert’s axiomatic treatment of geometry. Here, the purely mathematical part of knowledge is separated from geometry (where geometry is thought of as the science of spatial figures) and is then investigated directly.
Magal then uses abstract algebra to illustrate the point:
Again the key to the discussion is the question of content. When mathematics is viewed as a variant of logic it could easily be judged to have no specific content. The various arguments presented are complex, and not everyone writes with respect to the same logic. But the consistency of Bernays’ argument is most interesting to me. He is very clear on the question of content in mathematics. And reading this sent me back to another of his essays, where he is responding to Wittgenstein’s thoughts on the foundations of mathematics is 1959. Here he challenges Wittgenstein’s view with the nothingness of color.
Near the end of the essay he makes a reference to the Leibnizian conception of the characteristica universalis which, Bernays says was intended “to establish a concept-world which would make possible an understanding of all connections existing in reality. This dream of Leibniz’s (which it seems Gödel thought feasible) is probably the subject of another blog. But in closing I would make the following remarks:
Cognitive scientists have found that abstraction is fundamental to how the body builds meaning or brings structure to its world. This is true in visual processes where we find cells in the visual system that respond only to things like verticality, and it is seen in studies that show that a child’s maturing awareness seems to begin with simple abstractions. Mathematics is the powerful enigma that it is because it cuts right into the heart of how we see and how we find meaning.
I’d like today to stay on the topic of mathematics from the cognitive science perspective, and in particular, to make available another set of interesting studies summarized by C. R. Gallistel, Rochel Gelman and Sara Cordes. The studies are described in their contribution to the book Evolution and Culture (edited by Stephen C. Levinson and Pierre Jaisson and published by MIT press) and entitled: The Cultural and Evolutionary History of the Real Numbers. A pdf of this selection can be found here. These are provocative ideas that don’t seem to be getting a lot of attention yet.
Observations of the conceptual need for the real numbers, as well as their sometimes unwelcome presence, is peppered throughout the history of mathematics. But they were only formerly defined in the 19th century. The authors clarify that this real number system – a continuous, uncountable set of rational and irrational numbers
I liked their summary of the observed weakness of the rationals.
And this is the heart of my interest:
What I find provocative about the history of mathematics is while it may look like mathematics is just the conscious organization of practical symbols, over time it is inevitably discovered that these symbols contain more than was put into them. They grow deeper, become more entwined and produce unanticipated new possibilities. This has always suggested to me that every formalized idea emerges from a well-spring of possibilities to which the mathematician keeps gaining proximity. This alone is full of implications about the nature of abstract ideas, what they accomplish, and what moves the development of human culture. Recent papers, like this one on the evolutionary history of the real numbers, consistently encourage me to keep thinking along these lines.
The way these investigators identify the presence of this primitive use of continuous mental magnitudes is interesting. Some of the first studies cited involve pigeons, rats and monkeys, where their memory of ‘duration’ is observed by exploiting one of the difficulties with continuous measurements. The difference between nearby numbers is difficult to discern, for example, numerosities are represented by voltage levels, because of the noise in voltage levels. This is contrasted with numerosities represented by digital computers. Experiments were designed to identify one of these creature’s subjective judgment of durations, by using the behavior of the animals, as the indicator of their memory of duration. The variability in these judgments (called scalar variability) increases as the remembered durations get longer. It is believed that this is because the noise in a magnitude is proportional to the size of the magnitude. Their observations are fairly precise, and even extended to allow the observation of non-verbal animals doing arithmetic with these continuous magnitudes. Other studies designed to produce non-verbal counting in humans produced the same results. These mental magnitudes were also seen mediating judgments of the numerical ordering of symbolically presented integers.
I expect this kind of evidence will continue to grow.
For now I’ll leave you with their summaries:
I am increasingly fascinated by the mathematics of fundamental cognitive processes – like creatures finding their way to and from significant locations, or foraging for food, or foraging with the eyes, or comprehending the duration of an event. I’m excited by the fact that there are cognitive neuroscientists that have become focused on the architecture of these processes in particular. Their work seems to always suggest that our formal mathematical systems are growing out of these very same processes.
Gallistel is concerned with the abstractions of space, time, number, rate and probability that have been experimentally studied and found to be playing a fundamental role in the lives of nonverbal animals and preverbal humans. His premise is this:
He makes a point of distinguishing between magnitude and our symbolic numbers. Magnitudes are what he calls ‘computable numbers’ a quantity that “can be subjected to arithmetic manipulation in a physically realized system.”
Being a bit pressed for time, I’ll just reproduce some of his observations.
The representation of space, he says:
But in order to have a directive function, these representations of experienced locations must be vectors – ordered sets of magnitudes. And the organism accomplishes arithmetic with them.
Gallistel challenges the notion that time-interval experience is generated by an interval-timing mechanism, pointing out that
Instead, he proposes
I found one of the most interesting parts of this discussion to be the one on closure.
I find this particularly interesting because it took us some time to find signed differences in our symbolic system of subtraction or even to recognize the significance of closure.
I’ll end this with his brief conclusion. Some of the details of these studies can be found in the linked pdf.
A recent blog from Jennifer Ouellette (from the Scientific American Blog Network) brought my attention once again to how mathematics is related to the structure-building functions of the brain. As I followed up on some of the references in her post, I found myself on a little journey through hallucinatory experiences that I really enjoyed.
Her post is generally about Turing models applied to patterns found in the characteristic features of animals. But she got into territory that I find particularly provocative when she began to discuss the evidence for whether a Turing model can be applied to neurons in the brain.
I did a blog about how Turing insights appear to bridge otherwise disparate trends in science. But the link to Turing in this context is drawn through hallucination patterns, categorized in the early 20th century by University of Chicago neurologist Heinrich Kluever, into what he called the form constants: checkerboards, honeycombs, tunnels, spirals, and cobwebs.
The paper is very technical and involves quite a lot of mathematics as well as neuroscience. But the authors succeed in modeling a structure of the primary visual cortex whose activity would produce “certain basic types of geometric visual hallucinations.” Bressloff and Cowam conclude:
Apparently they found that the patterns predicted by their calculations closely matched what people will see when under the influence of hallucinogenic drugs, and they suspect that these patterns could be emerging from a kind of Turing mechanism.
Ouellette also spoke with neuroscientist Robin Carhart-Harris, who has done quite a lot of work on the brain mechanisms that lead to hallucinatory experiences, and how they might be used to help in the treatment of depression and addiction. I very much enjoyed watching an interview with him, shot as part of a forthcoming documentary on consciousness. There he made some really nice observations of the integrative function of ‘brain hubs’ that unify activity from different regions of the brain into coherent patterns or narratives. The geometric patterns of hallucinations are perceptual errors but, he tells us, the error “is a function of how the perceptual system works.” Carhart-Harris argues that the way we currently understand the action of the visual system, “says, very strongly, that reality is a construction. Reality only becomes something as we piece it together.”
With respect to how hallucinatory patterns reflect brain activity, Carhart-Harris tells Ouellette:
This is, in fact, what I have thought about mathematics.
I saw an opinion piece by Stephen Ornes, in the March 16 issue of New Scientist which ties the ongoing debate about the nature of mathematical ideas, to a modern one about money and ownership. Ornes argues that patentability is one of the most hotly contested issues in software development. The problem, as many see it, is that not all software is patentable because of its dependence on mathematics. Mathematics is understood as the exploration of abstract ideas, not the invention of new products. Ornes referred to an essay by David Edwards (University of Georgia) in the April 2013 issue of Notices of the American Mathematical Society. In the end, Edwards is calling for an update of the patent laws because the current laws do not promote the development of technological innovation. I wasn’t very inspired by the discussion. However, when I went to find the Edwards piece in the AMS Notices, I stumbled upon an essay, written in a completely different spirit, and published in January 2012. Jason Scott Nicholson, then a Ph.D. candidate in mathematics at the University of Calgary, addressed Eugene Wigner’s consistently cited query into the “unreasonable effectiveness of mathematics.” But Nicholson explores the puzzle of mathematics’ effectiveness using the structure of the ideas brought to life in the book Lila by Robert M. Persig, author of the widely read Zen and the Art of Motorcycle Maintenance.
Nicholson explains that, in Lila, reality is dual-aspected. One of these aspects is what Persig calls Static Quality, and the other, what he calls Dynamic Quality. A very brief explanation of these ideas is this:
Dynamic Quality is understood as the creative urge, the constant stimulus to move, perhaps to something ‘better.’ Static Quality is what is given in the patterns reflecting the “realization” of the undefined Quality that is the world. Static Quality is created in response to Dynamic Quality. It exists on 4 discrete but related levels:
Inorganic Biological Social and Intellectual
In this system, the biological builds on the inorganic, the social on the biological, and the intellectual on the social. Nicholson tells us that Perig uses a computer analogy to illustrate this idea:
Persig’s Static Quality creates a relationship among manifold patterns – from the bonding of atoms, to the mating of animals, to the formation of nations, to the dogma of religions, and the intellectual patterns of art and science. And this relatedness becomes the crux of Nicholson’s argument:
This configuration of Quality, Dynamic Quality and Static Quality is also used by Nicholson to describe the art/science character of mathematics:
The nature of Pirsig’s ‘Quality,’ and the use that Nicholson makes of it, reminded me of Leibniz again. For both Pirsig and Leibniz, our perceived reality is the consequence of structure being brought to something we cannot see, something that isn’t even material in the way we understand material. For Leibniz, this fundamental reality is the harmonious existence of monads. Leibniz’s monad is:
The text of Leibniz’s Monadology is not easy reading. It is a heavily logic-based analysis. The Internet Encyclopedia of Philosophy is one of many philosophy sites that discusses the document. There the point is clarified that:
There is no formal correspondence between the Persig and Leibniz. But there are most certainly parallels. Leibniz’s appetitions, for example, as explained by the Stanford Encyclopedia of Philosophy are:
I’ve been intrigued for some time by the view of reality Leibniz gave us and, to a large extent, because of its unmistakeable mathematical character. But I’ve also been captivated by how non-materialistic it is. Also from The Stanford Encyclopedia of Philosophy is this about Leibniz’s philosophy of mind.
I thought it worthwhile to bring these ideas up again in the context of Jason Scott Nicholson’s response to Wigner.
A nice article, focused on the origins of creativity, appears in the March 13 issue of Scientific American. Author, Heather Pringle, surveys research that seems to indicate that the human talent for innovation actually emerged over hundreds of thousands of years ago, before homo sapiens left Africa. This is contrary to the view held previously that a genetic mutation ignited sudden cognitive advances in homo sapiens already living on the European continent some 40,000 years ago.
Pringle describes the archeological evidence that motivates this revision in the story of our evolution. She also fills the story in with new insights into the evolution of the modern human brain. While mathematics is never mentioned specifically, questions about the emergence of symbol in our experience are inevitably relevant to questions about the source of mathematical creativity.
I followed up on one of Pringle’s examples, finds at an archeological site on the very tip of Africa.
I have given most of my attention to the engraved patterns on ochre. Professor Christopher Henshilwood led the early investigation at this site. His work continues more recently through the TRACSYMBOLS Project. A wealth of information related to the project is easily accessed on their website. Their home page has two videos and, in one of them, Ian Tattersall, Curator of the Museum of Natural History, introduces the work with a description of symbolic thought that certainly calls mathematics to mind (at least it does to my mind).
This last statement is indicative of the trend in cognitive neuroscience to put us back into the larger history of our lives.
Science News had an article on the finds at Blombos Cave in June, 2009. Here, author Bruce Bower, refers to the ochre engravings found in Blombos Cave as ‘meaningful geometric designs.’ The article can be accessed on this website.
An earlier (2007) paper in the Journal of Archeological Science focuses on another African excavation site on the western cape. But in that paper Alex Mackay and Aara Welz define the terms of the debate, namely, what we consider ‘design,’ and how we understand ‘symbolic.’ I thought these observations worth including.
Pringle completes her Scientific American piece with references to anthropological studies of the brain and the effect that the size of a population has on the likelihood that new ideas will emerge from with it and connect with other ideas that enhance their utility. Regarding the brain,
Researchers have imagined that it is this bigger brain that led to our ability to free-associate, and also to encode finer-grained memory. But free association needs analytic thought (referred to as the default mode) if we are to make something of freely associated connections. The body’s somehow learning to regulate subtly altering concentrations of dopamine (and other neurotransmitters) in order to switch smoothly from one mode to another, may be one of the keys to our idea-driven modern lives. And this mechanism, they say, could have taken tens of thousands of years to fine tune. These ideas are now being tested on an artificial neural network. Pringle concludes:
I enjoyed this view of our ancestors. And if our talent for free association is as critical to creativity as it would seem, one might wonder about how this electrochemical move from thought to thought actually translates into a useful idea. I think this question runs parallel to many of our questions about how surprisingly effective mathematics can be in the exploration of our worlds.
I read another New Scientist article today. The article was written by Brian Greene. While it didn’t give me a lot of new information, it made an interesting point about what it means (and when is it particularly effective) to take our mathematics seriously. He talked about Einstein’s insight regarding the speed of light. It was in the late 1800s, he explains, when Maxwell’s equations gave it the value of 300,000 kilometers per second (close to experimental measurements). But the equations didn’t say anything about the standard of rest that gave this speed meaning. Greene reminds us of the postulated invisible medium for transmitting light (the ether) which he calls a makeshift resolution to the problem. He then goes on to highlight a particular aspect of Einstein’s insight.
This is a detail about Einstein’s thinking that I hadn’t understood in quite that way. It’s a provocative idea. Mathematical necessity overrides the expectations created by our physical intuition. If the equation doesn’t depend on a standard of rest, than neither does the speed of light. Mathematics, here, is acting much like a human sense, a mode of perception.
After reading this, my own thoughts went down a number of different paths, which I can’t recall well enough to repeat here. But the precedence that mathematics has taken in physical theories, eventually led me to look at discussions centered around whether reality was fundamentally made of material or meaning. One of the schools of thought that reflects this question finds information to be more fundamental to reality than material. Paul Davies and Niels Henrik Gregersen compiled a collection of essays that address this issue in the book Information and the Nature of Reality. In his introduction, Davies describes Einstein’s theory of special relativity and general relativity as the first blow to our confidence in the idea of ‘matter.’
Later, of course, quantum physics not only amplified this question, but also raised other yet unanswered questions about the significance of the observer. Again from Davies:
In an interview for the radio show To The Best of Our Knowledge Davies said this about the view of reality that quantum theory may be expressing:
Certainly this opens the door to theological discussions, which the book does include. But just as interesting is the more fundamental question: What is the mode of perception that mathematics provides? Our visual systems structure the data that floods the retina. To what does mathematics give structure?