I recently listened to Krys Boyd’s interview with Peter Mendelsund, author of the new book What We See When We Read, on North Texas’ public radio. Mendelsund is an award-winning book jacket designer. The interview had the effect of connecting his thoughts about reading to thoughts that I have had about mathematics. It wasn’t immediately obvious, even to me, why. But I think I’m beginning to understand.
An excerpt from the book was published in the Paris Review. This excerpt focuses on the incompleteness of the visual images that our minds create when we are reading, despite the fact that we experience them as clear or vivid. Mendelsund quotes William Gass who wrote on the character of Mr. Cashmore from Henry James’s The Awkward Age:
We can imagine any number of other sentences about Mr. Cashmore added … now the question is: what is Mr. Cashmore? Here is the answer I shall give: Mr. Cashmore is (1) a noise, (2) a proper name, (3) a complex system of ideas, (4) a controlling perception, (5) an instrument of verbal organization, (6) a pretended mode of referring, and (7) a source of verbal energy.
The quote is from the book Fiction and the Figures of Life, a collection of essays first published in 1979. Following Gass a little further we find these remarks:
But Mr. Cashmore is not a person. He is not an object of perception, and nothing whatever that is appropriate to persons can be correctly said of him. There is no path from idea to sense (this is Descartes’ argument in reverse), and no amount of careful elaboration of Mr. Cashmore’s single eyeglass, his upper lip or jauntiness is going to enable us to see him.
Mendelsund adds this:
It is how characters behave, in relation to everyone and everything in their fictional, delineated world, that ultimately matters…
Though we may think of characters as visible, they are more like a set of rules that determines a particular outcome. A character’s physical attributes may be ornamental, but their features can also contribute to their meaning.
(What is the difference between seeing and understanding?)
He follows this with a very mathematical looking statement where the characters (along with some physical attributes), as well as particular events and their cultural environment, are represented by letters. Their interaction is somehow formalized in symbol.
These are all words that have been used with respect to mathematics – “not an object of perception,” “behavior that matters only in relation,” “a set if rules that determines a particular outcome…”
Mendelsund occasionally uses mathematical ideas to describe some of what may be happening in the reading (and the writing) of a story. There are the maps of novels, the graphs and contours of plot, the vectors in Kafka’s vision of New York City. And these observations:
Anna can be described as several discrete points (her hands are small; her hair is dark and curly) or through a function (Anna is graceful)
If we don’t have pictures in our minds when we read, then it is the interaction of ideas – the intermingling of abstract relationships – that catalyzes feeling in us readers. This sounds like a fairly unenjoyable experience, but, in truth, this is also what happens when we listen to music. This relational, nonrepresentational calculus is where some of the deepest beauty in art is found. Not in mental pictures of things but i the play of elements…
…But we don’t see “meaning.” Not is the way that we see apples or horses…
Words are like arrows – they are something and they also point toward something.
Any text can be seen as communication through words (symbol), that can be aided by pictures, but that only lightly relies on them. The reader builds an internally consistent world, grounded mainly in concepts, whose structure is communicated in symbol. And both structure and meaning are never fully completed. This certainly sounds a lot like mathematics. But more striking about Mendelsund’s work in particular, is his making direct use of his experience to explore profound philosophical questions. What happens when we read tells us something about ourselves.
The world, as we read it, is made of fragments. Discontinuous points – discrete and dispersed.
(So are we. So too our coworkers; spouses; parents; children; friends..)
We know ourselves and those around us by our reading of them, by the epithets we have given them, by their metaphors, synechdoches, metonymies. Even those we love most in the world. We read them in their fragments and substitutions.
The world for us is a work in progress. And what we understand of it we understand by cobbling these pieces together – synthesizing them over time.
It is the synthesis that we know. (It is all we know.)
And all the while we are committed to believing in the totality – the fiction of seeing.
…Authors are curators of experience.
…reading mirrors the procedure by which we acquaint ourselves with the world. It is not that our narratives necessarily tell us something true about the world (though they might), but rather that the practice of reading feels like, and is like, consciousness itself; imperfect; partial; hazy; co-creative.
Writers reduce when they write, and readers reduce when they read. The brain itself is built to reduce, replace, emblemize…Verisimilitude is not only a false idol, but also an unattainable goal. So we reduce. And it is not without reverence that we reduce. This is how we apprehend the world. This is what humans do.
Picturing stories is making reductions. Through reductions, we create meaning.
There is significant overlap here with how I see the doing and the making of mathematics. Mathematics is the making of meaning through reduction and synthesis. Emerging from some adjustment in the direction of the mind’s eye, mathematics mirrors, in another way, how we are acquainted with the world. It finds meaning that opens up other parts of that world, a bit more for us. And it tells us something about the nature of vision and understanding itself. Mathematics will not be fully embraced by our culture until we see this – until we recognize its own living nature.
The CogSci 2014 Proceedings have been posted and there are a number of links to interesting papers.
Here are some math-related investigations:
A neural network model of learning mathematical equivalence
The Psychophysics of Algebra Expertise: Mathematics Perceptual Learning Interventions Produce Durable Encoding Changes
Two Plus Three is Five: Discovering Efficient Addition Strategies without Metacognition
Modeling probability knowledge and choice in decisions from experience
Simplicity and Goodness-of-fit in Explanation: The Case of Intuitive Curve-Fitting
Cutting In Line: Discontinuities in the Use of Large Numbers by Adults
Applying Math onto Mechanism: Investigating the Relationship Between Mechanistic and Mathematical Understanding
Pierced by the number line: Integers are associated with back-to-front sagittal space
Equations Are Effects: Using Causal Contrasts to Support Algebra Learning
One of the presentations I attended is represented by the paper:
Are Fractions Natural Numbers Too? This study challenges the argument that human cortical structures are ill-suited for processing fractions, a view which has been used to justify the well-documented difficulty that many children have with learning fractions.
Such accounts argue that the cognitive system for processing number, the approximate number system (ANS), is fundamentally designed to deal with discrete numerosities that map onto whole number values. Therefore, according to innate constraints theorists, fractions and rational number concepts are difficult because they lack an intuitive basis and must instead be built from systems originally developed to support whole number understanding.
…Emerging data from developmental psychology and neuroscience suggest that an intuitive (perhaps native) perceptually based cognitive system for grounding fraction knowledge may indeed exist. This cognitive system seems to represent and process amodal magnitudes of non- symbolic ratios (such as the relative length of two lines).
This particular study is fairly well-focused, however. Researchers aim to demonstrate a link between our sensitivity to non-symbolic ratios and the acquired understanding of magnitudes represented by symbolic fractions. Given this focus, the study looked at individual responses to “cross-format comparisons of various fractional values (i.e. ratios composed of dots or circles vs. traditional fraction symbols). For example, a given symbolic ratio was given with a numerical numerator and a numerical denominator. The dot stimulus would show an array of dots in the numerator, of a certain quantity, and another array, of a different quantity, in the demoninator. The circle stimulus showed a blackened disc of a certain area in the numerator and another, of a different area, in the denominator. With a reasonable amount of care taken in their analysis, the authors concluded that they had found evidence of “flexible and accurate processing of non-symbolic fractional magnitudes in ways similar to ANS processing of discrete numerosities.
Considered in concert with other recent findings, our evidence suggests that humans may have an intuitive “sense” of ratio magnitudes that may be as compatible with our cortical machinery as is the “sense” of natural number. Just as the ANS allows us to perceive the magnitudes of discrete numerosities, this ratio sense provides humans with an intuitive feel for non-integer magnitudes.
An important consequence of this kind of evidence is their suggestion that the widespread difficulty with fractions may be the result teaching fractions incorrectly - with partitioning or sharing ideas that use counting skills and whole number magnitudes instead of encouraging the use of what may be our intuitive ratio processing system. This point was driven home for me when I looked at the circle representations of ratios that were used in the study. I found them very effective, very readable.
This view is certainly consistent with the proposal that a mental representation of continuous magnitudes predates discrete counting numbers (as with Gallistel, et al). But I also think that this initiative points to something likely to be important in cognitive science as well as math education. My own hunch is that an intuitive sense of ratio is likely grounded in continuous magnitudes, like length and area or perhaps even in tactile sensations of measure, like in cooking, as was suggested by one of the paper’s authors. And I think it plays some role in the long debate over the relationship between discrete and continuous numbers that can be seen in the history of mathematics. One could argue that the ancient Greek’s rigorous distinction between number and magnitude contributed to their remarkable development of geometric ideas. With the number concept isolated away from the geometric idea of magnitude, perhaps their geometric efforts were liberated, allowing a focused elaboration on that ‘intuitive sense of ratio,’ extending it, permitting manifold and deep results. Understanding this cultural event in the light of cognitive processes might inform our ideas about how mathematics emerges as well as how to communicate that development in mathematics education.
Yesterday I gave a talk at a symposium at the 36th annual Cognitive Science Conference. The content of the talk was described this way in our symposium proposal:
Mathematics has been the subject of experimental studies in cognitive science that explore the sensory grounding of number and magnitude. But mathematics also provides conceptual schemes that can manage our comprehension of complex integrated neural activity, like Giulio Tononi’s qualia space. Visual processes, like stereopsis, may be said to be mathematical in character, and the brain is often described as performing computations on sensory data as it constructs the elements of our experience. Mathematician Yehuda Rav has argued that mathematics grows on the scaffolding of cognitive mechanisms that have become genetically fixed with human adaptation. Joselle Kehoe will present a philosophy of mathematics informed by the significance of selected studies in cognitive science and selected moments in the history of mathematics. It will be considered in the light of structural coupling – the embodiment concept of enaction introduced by Varela, Thompson and Rosch.
I had the opportunity to listen to a number of talks and was struck by the extent to which computer modeling governs investigative strategies. These models are designed to mirror neural processing. The models inevitably influence the formulation of new questions, which often provide a refreshing angle on the phenomenon being investigated. But I began to wonder if modeling strategies might not also begin to create conceptual paths from which it may become difficult to exit. It also became increasingly clear to me that modeling of this sort can capture mathematical relationships (like a particular differential equations) in such a way that the formal mathematical description of the relationships could appear to not be necessary. The impulse from some to dismiss the need for formal representations could raise some interesting questions about the nature and the role of these representations. They certainly help identify the paths that connect concepts within mathematics? And these bridges between different branches of mathematics powerfully extend our understanding of both the physical and the conceptual. While many of the things I heard discussed were not designed to talk about what mathematics is, they indirectly addressed exactly that question.
A brief summary of my own talk, with links to supporting content, is this:
A view, held by many in cognitive science, is that an innate sense of magnitude forms a non-symbolic real number system. These are continuous magnitudes that arise from our experience of time, space, motion, etc. This view is supported by the observation that pre-verbal humans and non-verbal animals do perform non-symbolic arithmetic operations on these magnitudes. There are also studies that address how even plants are able to adjust their rate of starch consumption based on the amount of starch they have stored, and the length of time till dawn (when they can begin storing again). Researchers have argued that language, somehow, picked out the discrete integers with which we learn to count (since language itself is a discreet system). The evolution of the real number system in mathematics could then be seen as an investigation of the relationship between number and magnitude. From the Greek separation of quantity (number) and magnitude (the ratios and proportions of continuous quantities like length and time), through the 18th century difficulties with infinitesimals in calculus, and the eventual definitive placement of irrational numbers on the real number line, mathematicians continued to test these relationships and their implications. It is as if this work reflects a struggle to talk about a number system that existed before we were able to talk.
Brain processing itself seems to have a mathematical character – particular neurons in the visual system will only respond to particular orientations (a working abstraction), and optical illusions follow the pattern of statistical judgments (what we see is the most likely interpretation of the retinal image we have). These probabilistic inferences are also used to model learning in a fairly general sense.
Riemann’s 1854 work on the foundations of geometry can be seen as an instance of a mathematician’s point of view being informed by ideas about perception. We know this only because he acknowledged the influence of John Friedrich Herbart in his famous paper on the foundations of geometry. Herbart held the view that space was not the thing that contained the objects around us, but rather a mental image constructed by any number of things seen in relation to each other (including time and color). The combined influence of Gauss and Herbart moved Riemann’s thinking to propose that the concepts of measure and geometry could only have meaning if they began with the most general idea, a manifold, – any collection of elements that were related either discretely or continuously. This work is key to Relativity and points to both set theory and topology.
Mathematics has become increasing significant in the sciences. David Deutsch’s recent work on Constructor theory can be used as an illustration of a mathematics (an algebra in this case) structured to produce, within itself, new statements that cannot be expressed in current physical theories but have new physical theoretical content.
The point of these observations is to suggest that mathematics has played the dual role of characterizing what we see as well as how we see it. If we imagine that an effective body/brain would have a structure that somehow maps with, or couples with its world, then a very interesting question arises – how are we matched? Mathematics may be in the best position to contribute an answer to this question. It may be able to tell us something new about both ourselves, and about how the ‘thoughtful’ is found in nature.
In this way, mathematics is an inquiry like both art and science, looking at our experience and all that we see around us. Perhaps it can be used to refresh our view of both.
As I have investigated all of the things in science and mathematics that get my attention, I have developed an impression of mathematics that, philosophically, seems most consistent with Humberto Maturana’s biology of language. Maturana outlines his perspective in great detail in an essay by the same name that appeared in 1978 in the text Psychology and Biology of Language and Thought, edited by George Miller and Elizabeth Lenneberg.
Language must arise as a result of something else that does not require denotation for its establishment, but that gives rise to language with all its implications as a trivial necessary result. This fundamental process is ontogenic structural coupling, which results in the establishment of a consensual domain.
In a piece that appeared in Cybernetics and Human Knowing, Maturana says the following:
Living in language, doing all the things that we do in language, however abstract they may seem, does not violate our structural determinism in general, nor our condition as structure determined systems. As I observed our languaging behavior and the behavior of other animals, I realized that the central aspect of languaging was the flow in living together in recursive coordinations of behaviors or doings, and that notions of communication and symbolization are secondary to actually existing in language.
Language cannot be understood as a biological phenomenon if we do not take seriously our operation as structure determined systems. If we do not do so we remain trapped in the belief that language is a system of communication and thinking with representations (symbolizations) of an independent reality that contains us as its primary constitutive feature. And if we do not understand language as a biological phenomenon we shall remain in the mystery of self-consciousness through believing that this somehow reveals an intrinsic cosmic duality, and we shall not be able to understand ourselves as the self-conscious transitory beings that we are.
I see at least two important ideas that are needed to bear the weight of Maturana’s perspective. The first of these is the dynamic interaction of unities:
…we human beings exist in structural coupling with other living and not living entities that compose the biosphere in the dimensions in which we are components of the biosphere, and we operate in language as our manner of being as we live in the present, in the flow of our interactions, in our domains of structural coupling.
The other is the shift away from questions about ‘being’ to an inquiry into ‘doing.’
As a result of this fundamental conceptual change, my central theme as a biologist (and philosopher) became the explanation of the experience of cognition rather than reality, because reality is an explanatory notion invented to explain the experience of cognition (see Maturana, 1980).
While there is no mention of mathematics in any of the Maturana pieces that I’ve looked at, it seems to me that the evidence is growing that mathematics happens, like language, and that we live in it, much like the way Maturana describes our “living in language.” And I suspect that what we might not yet understand is the extent to which we share it, as it exists in manifold forms throughout nature. Brain processing can look very mathematical (the abstractions in visual processing and learning, the Bayesian models of learning, etc.). Foraging patterns, eye movement patterns and the patterns in how we search for words all look the same. And, of course, our perception of the mathematical nature of the universe itself continues to enable an almost incomprehensible expansion of what we can know.
In a paper on How Humberto Maturana’s Biology of Cognition Can Revive the Language Sciences, Alexander Kravchenko takes note of some of the resistance to Maturana’s perspective but concludes with the following remarks:
One of the most important consequences of adopting the biology of language is the relational turn in approaching the mind/language problem. Much of what an organism does and experiences is centered not on the organism but on events in its relational experiential domain, one that crosses the boundary of skin and skull. In its endeavor to answer the question “How does the brain compute the mind?” the neural theory of language overlooks the incoherence of the proposition that mind is a complex computational function of the brain. In the biology of cognition there is no such thing as “the mind” in the operation of the nervous system, and “the mind” is nothing but an explanatory notion: “language, self-consciousness and mindedness are different forms of existing in the relational domain in which a living being lives, not manners of operation of the nervous system” (Maturana, Mpodozis & Letelier 1995: 25). (emphasis added)
It is this relational idea that can have a significant impact on a philosophy of mathematics, as it will inevitably locate mathematics both in and around us.
I will be joining a few colleagues for a symposium at CogSci2014 and I’ve been gathering some notes for my talk. The talk will focus on the impact of embodiment theories on a philosophy of mathematics. As I looked again at some of the things I’ve chosen to highlight in my blogs, I came upon a talk given by Josh Tenenbaum, Professor in the Department of Brain and Cognitive Sciences at MIT. I’ve read about aspects of his work before, but after listening to the talk he prepared for the Simons Foundation, I became even more interested in the implications that his investigation of Bayesian models of cognition might have for mathematics. One of the goals of the Simons Foundation talk was to highlight what Tenenbaum called “some of the new and deep math that has come out of the quest to understand intelligence.” I was particularly struck by his straightforward suggestion that statistics is not only intuitive, but that it may be part of our intuition.
The following remarks appear in the text introduction to the talk:
The mind and brain can be thought of as computational systems — but what kinds of computations do they carry out, and what kinds of mathematics can best characterize these computations? The last sixty years have seen several prominent proposals: the mind/brain should be viewed as a logic engine, or a probability engine, or a high-dimensional vector processor, or a nonlinear dynamical system. Yet none of these proposals appears satisfying on its own. The most important lessons learned concern the central role of mathematics in bridging different perspectives and levels of analysis — different views of the mind, or how the mind and the brain relate — and the need to integrate traditionally disparate branches of mathematics and paradigms of computation in order to build these bridges.
…The recent development of probabilistic programs offers a way to combine the expressiveness of symbolic logic for representing abstract and composable knowledge with the capacity of probability theory to support useful inferences and decisions from incomplete and noisy data. Probabilistic programs let us build the first quantitatively predictive mathematical models of core capacities of human common-sense thinking: intuitive physics and intuitive psychology, or how people reason about the dynamics of objects and infer the mental states of others from their behavior.
There are a few themes in this talk that are worth noting. There is certainly the suggestion that the brain’s computations are a kind of mathematics that happens within the body itself. Tenenbaum was clear, however, about the extent to which brain processes are not understood. No one yet knows how the brain actually learns, or how it translates symbols into meaning, nor how any of this can be understood with respect to the activity of neurons. But he has been gathering evidence, on more than one front, that supports the idea that probability theory has a lot to say about “the things that the brain is good at” — like visual perceptions, learning the cause and effect relationships in the physical world, understanding words and the meaning of actions…etc.
In one of the studies Tenenbaum discussed, Bayesian estimates that were computed from priors (defined by empirical statistics) were compared to the judgments that individuals made when asked to make predictions about those same variables. Subjects in the study were asked to estimate things like the amount of money a movie will gross, a human life expectancy, or a movie’s run time. For example, they might be asked, if you read about a movie that has made $60 million to date, how much will it make in total. There were five groups of subjects, and each of the groups were given different numbers. In our example one group may have been given $30 million, another $50 million, etc. The prior for a parameter describes what is known, a priori, about the parameter being estimated. In this case, what is known was gathered from empirical data. And the distribution of priors for the various categories (movies runs, life expectancies, movie grosses) are qualitatively different from each other. The data for one of the variables, for example, may produce a power distribution, while the data for another produces a Gaussian distribution. You can apply Baye’s rule to each of these distributions to compute a posterior distribution – priors updated by experience or evidence. The median of the posterior distribution provides what’s called a posterior predictive estimate. In the studies Tenenbaum cites, applying Bayesian inference to the priors fits very well with the estimates people actually made, both quantitatively and qualitatively.
A recent paper entitled A tutorial introduction to Bayesian models of cognitive development, which Tenenbaum co-authored, is a broad treatment of how and why Bayesian inference is used in probabilistic models of cognitive development. There, the point is made that the Bayesian framework is generative. By generative is meant that the data observed has been generated by some underlying process. And one of the values of Bayesian models is their flexibility.
…because a Bayesian model can be defined for any well-specified generative framework, inference can operate over any representation that can be specified by a generative process. This includes, among other possibilities, probability distributions in a space (appropriate for phonemes as clusters in phonetic space); directed graphical models (appropriate for causal reasoning); abstract structures including taxonomies (appropriate for some aspects of conceptual structure); objects as sets of features (appropriate for categorization and object understanding); word frequency counts (convenient for some types of semantic representation); grammars (appropriate for syntax); argument structure frames (appropriate for verb knowledge); Markov models (appropriate for action planning or part-of-speech tagging); and even logical rules (appropriate for some aspects of conceptual knowledge).
The representational flexibility of Bayesian models allows us to move beyond some of the traditional dichotomies that have shaped decades of research in cognitive development: structured knowledge vs. probabilistic learning (but not both), or innate structured knowledge vs. learned unstructured knowledge (but not the possibility of knowledge that is both learned and structured)p10
In his talk, Tenenbaum says that intelligence is about finding structure in data. This is key, I think, to why mathematics has played so prominent a role in physics. And, as Tenenbaum says, “it is the math, not a body of empirical phenomena that supports reduction and bridge-building.” His talk for the Simons Foundation is aimed at highlighting the significance of mathematics in the study of cognition — from Bayesian inference to Bayesian networks (probabilistic graphical models) where arrows describe probabilistic dependencies and algorithms compute inferences. His arguments lead to the development of probabilistic programming. He also made a brief reference to mathematics being explored by some of his colleagues, that would serve to translate inferences into stochastic (random) circuits, suggesting a potential parallelism to the brain.
Neuroscience, Tenenbaum points out, is using the language of electrical engineering and common sense is at the heart of human intelligence. He goes on to explain that the toolkit of graphical models is not enough to capture the causal processes underlying our intuitive reasoning about the physical world or our intuitive psychological/social reasoning. To do this, one needs to define probabilities “not over a fixed number of variables, but over something much more like a program.” Current work is focused on designing programs that can be run forward for prediction and backward for inference, explanation and learning.
You can find a nice account of recent work on probabilistic programing on Radar.
In Tenenbaum’s tutorial paper, he quoted Laplace (1816) who sort of summed things up when he said:
Probability theory is nothing but common sense reduced to calculation.
I listened to a couple of interviews with Gregory Chaitin on the Closer to Truth website. They may have been part of TV episodes that I haven’t seen but I was actually invigorated by some of the things he said, and it made me want to share them.
One of the interviews (in two parts) is found under the heading Is Mathematics Eternal, and is contained within the topic What are the Deep Laws of Nature? Here, Chaitin seemed most intent on dispelling the standard view of mathematics, where mathematics is seen, perhaps, as the epitome of rationality, or as thought without feeling, or logical structure without any inherent meaning. The discussion begins with a discussion of beauty.
We mathematicians are not machines,” he says, “we’re not calculating machines at all. We’re human beings and we’re emotional. Why are we devoting our lives to mathematics?
His reply is essentially that mathematics seduces them, that mathematicians are passionate about mathematics. He tells us that what many mathematicians are really looking for is beauty. And here he uses a phrase that is likely filled with a number of thoughts worth exploring. “It’s something sensual,” he says, “it’s the sensuality of ideas.” The whole body is somehow participating. And I believe that too, but how does one understand the sensuality of ideas? What does it say about how mathematics is happening. This sensuality may only be reached when one participates in mathematics at a certain depth.
Chaitin compared the effect that pure mathematics had on him as an adolescent boy, to the effect that women had on him. “It was confused in my case,” he tells us. The reason for the confusion, as he sees it, may be that beauty is in some way connected to the life force. “One wants to create something beautiful, one wants to be illuminated by it.” Chaitin uses the word illuminated more than once, and it is a particularly good choice of word.
When asked about what makes a proof beautiful, one of the things he said is, “at first it seems surprising and then it seems inevitable. And you ask yourself, how come you didn’t see it sooner?” Even when speaking about a beautiful painting, Chaitin uses the phrase, “it illuminates you,” like a light coming out of the canvas. He says the same of a beautiful proof, “it illuminates you.” I think the image (in both cases), that something becomes lit, is a provocative image, referring to both light and understanding. “A proof that isn’t illuminating is useless.” The purpose of a proof, he explains, is not to prove something, it’s to give you understanding. It should give you some intuition about why the result is right.
As the first segment nears its end, Chaitin brings up the mathematician Ramanujan.
I think Ramanujan is great because he contradicts everything that mathematics is supposed to be.
Chaitin tells us that Ramanujan believed that an equation is of value only if it expresses one of God’s thoughts. Ramanujan said of his own ideas that they were given to him, in his sleep, by the goddess Namagiri. Then Chaitin moves on to Cantor. “He is the most weird contradictory person.” Cantor was trying to understand God. He saw the contradictions inherent in his ideas but was not discouraged by them because, perhaps, his ideas were more important. Cantor was learning something about the infinite. Chaitin also reminds us that Poincare referred to Cantor’s worked as a disease from which he hoped future generations recovered. But Cantor’s work is foundational now. Having been cleansed of some of its difficulties, and with Cantor’s more irrational inspiration hidden, it strongly influenced the course mathematics took in the 20th century.
Chaitin is making the argument that these peculiar aspects of Ramanujan’s and Cantor’s experiences are ignored because they contradict the way people expect to think about mathematics.
I prefer to take the other extreme. I believe mathematics is based only on emotion and inspiration and it’s totally irrational and we don’t know where it comes from especially in cases where it’s an obsession.
Chaitin also recalled the work of Leonard Euler:
He created a lot of the mathematics that physicists and engineers use, but it was like a river, like a torrent of creative thought. Every week he would do another wonderful paper. He would give the whole train of thought and you would think when you read it, oh, I could do that. But no. Today a lot of his proofs are not considered rigorous. But even though his proofs are not what today is considered a valid proof, he discovered all this mathematics. Where did all those ideas come from?
In another segment, under the heading Is Information fundamental, Chaitin is addressing the question of whether information is more fundamental to the universe than matter. The inspiration for these ideas may be the computer. But what if the computer isn’t just a metaphor? Chaitin asks. A theory can be thought of as a computation. You input your theory and the output is the physical universe, or mathematical theorems. And when is a theory good? This goes back to Leibniz, he says. A theory is good when it’s a compression, when what you put into the computer is simpler or smaller than what you get out. Then you understand. And that understanding can be mathematical or it can be physical. And maybe the right way to think about the universe is that the universe is a computation – computing its future state from its current state. One can think of everything as a computation – understanding, the physical universe, DNA, as well as current technology.
Chaitin suggests that this provides a whole new way of looking at epistemology, which reach back to ideas presented by Leibniz after 300 years of development. We have reinvented an old question: Is the universe built of matter or of mind?
I wrote a bit on Leibniz a couple of years ago. I will revisit Leibniz and Chaitin for more on this.
Jason Padgett, author of the book Struck by Genius, appeared on CBS This Morning on April 24. On May 5, livescience also did a piece on him and his book. Padgett was assaulted in 2002 and suffered a severe concussion. But, following this head injury, he acquired an extraordinary facility for seeing mathematics. He is, as many say, an “acquired savant,” or prodigy in mathematics.
Padgett is quoted in the CBS article as saying:
I thought it was the pain medicine that they had given me that made me feel so strange….Things looked like individual picture frames coming in and clouds moving. Instead of looking smooth, they looked like little tangent lines in a spiral. Everything was discreet and chunky.
Padgett’s new skills are attributed, in part, to synesthesia, a condition where the senses blend – one might see music or hear color. From what I’ve read so far, it seems to be his visual experience that is most changed. But his visual experiences have motivated him to draw as well as do mathematics. In fact, for him, the mathematics appears to be contained in the drawings (which, in itself, is noteworthy).
Livescience author Tanya Lewis reports it this way:
With Padgett’s new vision came an astounding mathematical drawing ability. He started sketching circles made of overlapping triangles, which helped him understand the concept of pi, the ratio of a circle’s circumference to its diameter. There’s no such thing as a perfect circle, he said, which he knows because he can always see the edges of a polygon that approximates the circle.
According to the livescience article, functional magnetic resonance images of Padgett’s brain showed significant activation in the left parietal cortex, although it’s not clear in the article what Padgett is doing when the activation occurs. The parietal cortex is known to be an area where sensory information is integrated. I considered the relevance of the parietal cortex to mathematics in my +plus article as well as in a post from a few years ago.
In a 2005 article Edward Hubbard, Manuela Piazza, Philippe Pinel and Stanislas Dehaene write on the interactions between number and space in the parietal cortex. The authors conclude:
In the more distant future, it might become possible to study whether more advanced mathematical concepts that also relate numbers and space, such as Cartesian coordinates or the complex plane, rely on similar parietal brain circuitry. Our hypothesis is that those concepts, although they appear by cultural invention, were selected as useful mental tools because they fit well in the pre-existing architecture of our primate cerebral representations. In a nutshell, our brain organization both shapes and is shaped by the cultures in which we live.
But studies with Padgett went a bit further. In her article Lewis reports:
…the fMRI only showed what areas were active in Padgett’s brain. In order to show these particular areas were causing the man’s synesthesia, Brogaard’s team used transcranial magnetic stimulation (TMS), which involves zapping the brain with a magnetic pulse that activates or inhibits a specific region. When they zapped the parts of Padgett’s parietal cortex that had shown the greatest activity in the fMRI scans, it made his synesthesia fade or disappear, according to a study published in August 2013 in the journal Neurocase.
Berit Brogaard, who performed these studies, has shown in another study that “when neurons die, they release brain-signaling chemicals that can increase brain activity in surrounding areas.” The increased activity may not last, but if it produces structural changes, then the brain-activity changes would persist. Lewis doesn’t develop this idea but, for me, it calls to mind the kind of interaction ecologist Deborah Gordon described in her TED talk: What ants teach us about the brain, cancer and the internet. There Gordon describes how systems without a central control are directed by interaction. Ants operate within a network created by their interaction. For example, in situations where there may be a high cost to venturing out, the decision to forage for food can be a tricky one. Ants seem to add up the stimulation they receive from other ants to know whether they should begin. She compares the effectiveness of this interaction to neurons that add up the stimulation from other neurons to know whether to fire or not.
Author and psychiatrist Darold Treffert interviewed Padgett. Padgett writes:
…he told me that these innate skills are, in his words, “factory-installed software” or “genetic” memory. After interviewing me…..he also suggested that all of us have extraordinary skills just beneath the surface, much as birds innately know how to fly in a V-formation and fish know how to swim in a school.
While the experience of the savant is rare, and often coupled with disabilities of another kind, I think these mysterious talents provide a unique opportunity to explore the enigma of how we come to know anything. Perceiving very large quantities, in the absence of counting; having a sensory experience of numeric relationships; playing an instrument without ‘learning’ it, all these things happen. Treffert recounts some of his encounters in a very nice piece which appeared as a Scientific American Guest Blog last July. These remarkable abilities must be showing us something about the nature of the disciplines involved as well as the potential of brain functions.
In his book, Padgett says:
To me, a tree is more than its geometry, but geometry is also far more than most people realize. I think it’s everything.
And in the CBS report he says:
It means that we all have this ability within us,” he said. “I had no prior training whatsoever, and, like, we’re all doing complicated math and physics. … If somebody throws a ball and you catch it, you calculated the gravitational field, everything, and you caught that ball. Your brain- if you wrote that as an equation, that’s a serious equation to write, but you could just catch the ball automatically.
I recently considered the role that mathematics plays in bringing meaning, or perhaps even story, to our experience. Mathematics is often used to reveal the structure that can be found in large sets of data, or in any number of physical things that change over time, or in the properties of the abstractions themselves. Mathematics, then, sort of tells us what we’re looking at, within what would otherwise be an unwieldy or even meaningless data. In my last post, I took note of the way that mathematics was used to communicate meaning in the structure of cathedral walls – in particular Lincoln Cathedral whose construction began at the end of the 11th century and continued through the Middle Ages. There I shared the suggestion that the geometry of the Cathedral was used to convey, or even to instruct on, the fundamental nature of our being. It happens now that another New Scientist article, Shakespeare: poet, playwright, scientist broadened the issue for me.
The article was written by broadcaster and author Dan Falk, whose most recent book is The Science of Shakespeare. In the article, Falk considers, specifically, evidence in the play’s words that Shakespeare made use of astronomical observations, and explores how he may have known about them.
Galileo’s telescopic observations came toward the end of Shakespeare’s career, but Cymbeline, first performed in 1611, offers tantalizing hints that he was aware of the findings. In the play’s final act, the hero, Posthumus, falls into a dream-like state, and the ghosts of four family members appear and move around him in a circle. The ghosts cry out for Jupiter, the Roman god. On hearing their pleas, he descends onto the stage. So we have Jupiter and four ghosts moving in a circle.
But the book also addresses the emergence of the scientific world view more generally and, in doing so, takes note of the relationship between science and religion. I recently saw a group of students who had gathered on the University of Texas, Dallas campus with posters designed to provoke a debate about truth, science and religion. When I saw them, it occurred to me that mathematics never comes up in these debates, despite the fact that it is so intimately tied to ancient, medieval and modern world views. Reading the history of the relationship between science and religion, as Falk surveys it, encouraged me to pursue my own thoughts about whether an updated philosophy of mathematics might contribute to the science vs. religion debate. Early in his book, Falk makes the following observations:
…the rediscovery of classical texts, via Arabic translations, triggered a new wave of learning across Europe. Those works included the writings of Aristotle and Ptolemy…,as well as the geometry of Euclid, the medical writings of Galen, and much more…This wave of learning was closely linked to the activities of the Roman Catholic Church. The best medieval schools had been those associated with the monasteries and the great cathedrals. By the late Middle Ages these had also become centers of what we would now call science…There were also the universities, the earliest having been founded around 1200; these, too, functioned largely as religious institutions. The highest degree offered was in theology – though to obtain it, the student also had to master mathematics, logic, and natural philosophy.
The close connection between science and faith may seem strange to the modern reader, living at a time when Western society, and Western science in particular, has become a secular endeavor….The evolving relationship between science and religion is a large and complex subject…For one thing, religion was simply part of the fabric of society; all of the key figures of the Scientific Revolution were men of faith of one kind or another.
…As historian Paul Kocher puts it, early modern science “was more often cited as proving God’s existence than disproving it…And as Principe writes, the study of nature was seen as “an inherently religious activity…But the link between science and faith, as Principe stresses, is deeper than this. For the thinkers of early modern Europe, he writes, “the doctrines of Christianity were not personal choices. They had the status of natural or historical facts.”
Mathematics inspired many of the thoughts of an emerging scientific world-view, but it is also closely associated with the more mystical side of things. Kepler, as Falk points out, was obsessed with numerology, like the ancient Pythagoreans. His work relies on observations of the physical as well as the mystical. Quoting Allen Debus, Falk tells us:
His subtle mixture of mathematics and mysticism is “far removed from modern science, but it formed an essential ingredient of its birth.”
Mathematics is often at the beginning of things. The considerations of Shakespeare’s Hamlet explored in Falk’s article, and in his book, should remind us of the complexity of human thought. Relationships among art, science, and religion are inherent. They are all narratives that grow out of the same ground. Their coexistence, it would seem, is just a fact of our existence, and one of the keys to a deeper understanding of who we are.
If mathematics is, as I tend to see it, the mind itself building structure with the elements of thought, then mathematics’ development is like the development of another sense. It is, after all, the structure we give to any sensory data that creates a meaningful thing perceived. And so it is with mathematics. Mathematics, like story and name, brings meaning, not mechanics, to our experience. It increasingly links what we see to what we think we know, as we try to reconcile our immediate experience with what’s ‘out there.’ The evolution of mathematical concepts allows us to probe deeper and deeper into the universe, as well as into our own nature. And this is the subject of both religion and science. So why is mathematics so consistently absent from our debates?
In their March 22 issue, New Scientist reported on the recent detection of gravitational waves that are predicted by the inflationary theory in physics. This observation could help reveal details of what the cosmos was like “in the first slivers of a second” following the big bang. It supports the theory that implies the existence of an ever-expanding multiverse. There’s also a nice write up in the Guardian, which is free.
Lisa Grossman, the author of the report, followed up with a short piece (Medieval text about light held hints of a multiverse) describing a recent paper which explores the mathematical side of a 13th-century treatise on light written by Robert Grosseteste. This paper is one of the fruits of an interdisciplinary research project at Durham University, UK, which has been given the name The Ordered Universe. It’s core team of investigators includes Giles Gasper, from Durham’s Department of History, Tom McLeish from physics, Cecilia Panti from the University of Rome’s Department of Philosophy, and Hannah Smithson from the Department of Psychology at Pembroke College, University of Oxford, UK. This group along with Richard G. Bower, Brian Tanner and Neil Lewis, co-authored the paper in which Grosseteste’s treatise On Light (De luce) is reformulated in terms of modern mathematics. The authors conclude that Grosseteste’s ideas contain the prospect of the multiverse theories now being considered by cosmologists.
This kind of observation may help reveal something about the kinship between mathematics and thought itself. Grosseteste brought a conceptual order to non-mathematical thoughts. His own thoughts follow the patterns that he finds in his experience, in the thoughts of others, as well as in the very conceptual structures he considers. He follows the implications of the ideas he puts forward and then creatively interprets these implications.
Yet this non-mathematical order can accommodate the superpositioning of modern mathematics, out of which emerges something remarkably similar to contemporary theories in cosmology.
The abstract of the paper reads as follows:
In his treatise on light, written in about 1225, Robert Grosseteste describes a cosmological model in which the Universe is created in a big-bang like explosion and subsequent condensation. He postulates that the fundamental coupling of light and matter gives rises to the material body of the entire cosmos. Expansion is arrested when matter reaches a minimum density and subsequent emission of light from the outer region leads to compression and rarefaction of the inner bodily mass so as to create nine celestial spheres, with an imperfect residual core. In this paper we reformulate the Latin description in terms of a modern mathematical model. The equations which describe the coupling of light and matter are solved numerically, subject to initial conditions and critical criteria consistent with the text. Formation of a universe with a non-infinite number of perfected spheres is extremely sensitive to the initial conditions, the intensity of the light and the transparency of these spheres. In this “medieval multiverse”, only a small range of opacity and initial density profiles lead to a stable universe with nine perfected spheres. As in current cosmological thinking, the existence of Grosseteste’s universe relies on a very special combination of fundamental parameters.
It is this “very special combination of fundamental parameters,” where the nine-sphere universe depends on key initial conditions, that opens the door to a multiverse. Page two includes a careful description of the paper’s intent:
While it is crucial to avoid superposing a modern world view into Grosseteste’s thought, throughout his work there pervades an interest in the nature of the created world, the existence of order within it, the mechanisms whereby it is sustained and a search for unity of explanation. These ideas are common in medieval thinking; nonetheless the originality of Grosseteste was to think about unity, order and causal explanation of natural phenomena as being due to light, its properties and the mechanism by which we perceive it. …We are not trying to “correct” Grosseteste’s thinking in the light of modern physics, nor are we claiming Grosseteste’s ideas as a precedent for modern cosmological thinking. Rather, we are making a translation, not just from Latin into English but from the new critical Latin edition  and English translation  of his De luce into mathematical language. We aim to write down the equations, as he might have done had he access to modern mathematical and computational techniques, solve the equations numerically and explore the solutions. There are benefits here from both an historical and a scientific perspective. The application of mathematics and computation generate, as we shall see, a closer and more comprehensive examination of a medieval scientific text and the mind behind it. However, there are scientific benefits as well, as the medieval cosmos constitutes a quite novel arena to compute radiation/matter interactions and dynamics, and in which to discover new physical structure.
Grosseteste’s work builds on Aristotle’s idea that the earth is embedded in a series of nine concentric spheres that are the universe. Particularly striking, however, is Grosseteste’s idea that light is the “first corporeal form and that it multiplies itself infinitely.” He argued that light (luxe) expanded instantaneously from a point. For Grosseteste, this first corporeal form has no dimension and, since form and matter are inseparable, neither does matter. Only by its expansion into all directions does light introduce three dimensions into matter.
In the beginning of time, light extended matter, drawing it out along with itself into a sphere the size of the material universe.
Light drags matter outwards and so, Grosseteste concludes, the density of matter must decrease as its radius increases.
But finding Grosseteste’s cosmology in the architecture of Lincoln Cathedral makes things even more interesting. John Shannon Hendrix at Roger Williams University does just that in his 2010 paper, “The Geometries of Robert Grosseteste and the Architecture of Lincoln Cathedral.”
…the geometries were similar enough to suggest that a cultural concept of a geometrical substructure of matter could be translated into the architectural forms of the cathedrals as catechisms of the structure of matter and being. The architecture of the cathedrals was intended to edify the viewer as to the underlying nature of being and as to the relation between the human intellect and nature and God.
…Geometries used by Grosseteste to describe the diffusion and rarefaction of light in the formation of matter can be compared to the peculiar geometries of the vaulting of Lincoln Cathedral, in the particular lines and line segments of the vaulting. Volumes formed by the reflection and refraction of light, as described by Grosseteste, can be compared to the volumes of the vaulting, in particular the concave and conical shapes. The lux, or spiritual light, and the lumen, or physical light, can be applied to the light in the cathedral, as shining through the stained glass windows and illuminating the geometries. (emphasis added)
The Stanford Encyclopedia of Philosophy makes the following statements about Grosseteste:
Under Augustine’s influence Grosseteste subscribes to an illuminationist account of human knowledge, according to which human knowledge is understood by analogy to bodily vision: as a body can only be seen if light is shed on it and the eyes, so something can only be known if a spiritual light is shed on it and the mind’s eye. Grosseteste presents versions of such an account in his treatise On Truth and in his commentary on the Posterior Analytics.
…But as we can only see a body as colored if an external light is shed on its color, rendering it actually visible to us, so in order for us to see a created thing as true, an external light must be shed on its truth, rendering it actually visible to us. (emphasis added)
What we have here are a series of overlapping metaphors, if you like, each trying to capture (sensorily and intellectually) our own human impression of our existence and our origin. It may be said that mathematics brings us most precisely (in the intellectual sense) to the beginning of things, but mathematics, in all of them, is used to build and to communicate meaning, bringing meaningful structure to remote light signals, as well as cathedral walls. That it can speak to us about ourselves in so diverse a manner again indicates its fundamental nature in human experience.
A Scientific American article brought mathematical knitting to my attention once again, and within the article was a link to Bridges, an organization which oversees the annual Bridges conference that explores the connections between art and mathematics. Following the link to their 2013 Conference, I found their Short Movie Festival. I’ve watched a number of these short films whose actions explore things like topological objects, fractals, geometric symmetries, and groups. These ideas are presented in visual works, movement performances, as well as in nature. I thoroughly enjoyed all of them and I found myself asking the question, “So where does mathematics live, really?” For example, associated with the short movie Dancing Braids by Ester Dalvit is the following note:
Braids can be described as configuration space of points in a disc. These can be visualized as dances: the positions of each dancer are translated into a strand of the braid, the time into a spatial dimension.
This movie is a small part of a long video about braid theory which is available here.
Or with Susan Gerofsky’s film, The Geometry of Longsword Locks, is this:
In traditional English longsword dancing, a team of dancers makes intricate moves while joined together by their wooden or metal ‘swords’. An impressive element of the dance is the variety of traditional geometric, symmetrical sword locks (often stars) created through the movements of all the dancers. The film showcases a longsword dance and the locks created by the physical algorithms of the conjoined dancers’ movement. After showing the dance, questions are offered to spark mathematical explorations by secondary or post-secondary students. These questions include topological and geometric ideas about crossings, angles and edges, and logic-related questions about categorizing lock types and discovering whether new locks could be created through analysis of the physical algorithms that create them. Slow-motion and repeated views help learners explore this rich source of geometry.
On the Simons Foundation website is yet another short video on Change Ringing.
The art or “exercise” of change ringing is a kind of mathematical team sport dating from the 1600s. It originated in England but now is found all over the world. A band of ringers plays long sequences of permutations on a set of peal bells. Understanding the patterns so they can be played quickly from memory is an exact mental exercise which takes months for ringers to perfect. Composers of new sequences must understand the combinatorics of permutations, the physical constraints of heavy bells, and the long history of the art and its specialized vocabulary. Change ringing is a little-known but surprisingly rich and beautiful acoustical application of mathematics.
According to The North American Guild of Change Ringers,
the earliest record we have of these is from 1668:Tintinnalogia: or, The Art of Ringing. Wherein Is laid down plain and easie Rules for Ringing all sorts of Plain Changes. Together with Directions for Pricking and Ringing all Cross Peals; with a full Discovery of the Mystery and Grounds of each Peal.
Perhaps we can ignore the effect of the subject tabs we learned to put in our notebooks when we were young and ask some new questions. Do these visual and musical experiences represent mathematical concepts or are mathematical concepts actually exploring the elements of these visual and musical experiences? I lean in the direction of the latter. In fact, I would argue that one of the major functions of the brain is to integrate experience. The dances shown in two of the short films are, in some sense, an impulsive integration of the things we hear, see and hold, that become shapes within the inherent unity of our experience. It can be said that mathematics ‘picks up’ on this impulse, and further explores that unity by investigating the paths that are born of these more impulsive harmonies. Mathematics is then distinguished by its symbolic representation of the flow of patterns created by our living – by the visual, and audio structures that the senses build, as well as the cognitive structures that develop with them. Braids and knots are two of the oldest human impulses to create new experience, and they are two of newest objects investigated by mathematics, which then further integrates them into what we know of number and quantity and symmetry.
There is one more thing, not so much related to the theme of this post, but worth a look. One of the short movies in the Bridges short movie festival is a poetic approach to the words real and complex that I think is really nicely done. You can go to it directly here.