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Continuity, randomness and the Oracle

Flipping through some New Scientist issues from this past year, I was reminded of an article in their July 19 issue that brought together a discussion of the brain and mathematics with particular emphasis on the effectiveness of employing the sometimes counter-intuitive notion of the infinity of the real numbers. The content of the article, Know it all, by Michael Brooks, explores the viability of Alan Turing’s idea of the “oracle” – a computer that could decide undecidable problems. It highlights the work of Emmett Redd and Steven Younger of Missouri State University who think that they see a path to the development of this “super-Turing” computer that would also bring new insight into how the brain works.

The limitations on even the most sophisticated computing tools is essentially a consequence of limited power of logic. Mathematician Kurt Gödel’s Incompleteness Theorem shows clearly that any system of logical axioms will always contain unprovable statements. Turing made the same observation about a universal computer built on logic alone. Such a computer will inevitably come up against ‘undecidable’ problems, regardless of the amount of processor power available. But Turing did imagine something else.

…An oracle as Turing envisaged it was essentially a black box whose unspecified contents would be able to solve undecidable problems. An “O-machine,” he proposed, would exploit whatever was in this black box to go beyond the bounds of conventional human logic – and so surpass the abilities of every computer ever build.

Brooks then tells us about a computer scientist working on neural networks – circuits designed to mimic the human brain. Hava Siegelmann wanted to prove the limits of neural networks, despite their great flexibility.

In a neural net, many simple processors are wired together so that the output of once can act as the input of others. These inputs are weighted to have more or less influence, and the idea is that the network “talks” to itself, using its outputs to alter its input weightings until it is performing tasks optimally – in effect, learning as it goes along just as the brain does.

Siegelmann eventually observed an unexpected possibility. She showed that, in theory, if a network was weighted with the infinite, non-repeating numbers in the decimal expansion of irrational numbers such as pi, it could transgress the limitations of a universal computer built on logic alone. And this relies, it seems, on the generation of randomness produced by the irrational number.

While Siegelmann published her proof in 1995, it was not enthusiastically welcomed by fellow computer scientists.

…she soon lost interest too. “I believed it was mathematics only, and I wanted to do something practical,” she says. I turned down giving any more talks on super-Turing computation.

Ah, “mathematics only…,” she says.

Redd and Younger, aware of Siegelmann’s work, saw their own work headed in the same direction.

… In 2010, they were building neural networks using analogue inputs that, unlike the conventional digital code of 0 (current on) and 1 (current off), can take a whole range of values between fully off and fully on. There was more than a whiff of Siegelmann’s endless irrational numbers in there. “There is an infinite number of numbers between 0 and 1,” says Redd.

This infinity of numbers between 0 and 1, was one of the first things to intrigue me about mathematics. What are we looking at when we look at this infinity of numbers, whose size is the same as the infinity of the whole line?

In 2011 they approached Siegelmann, by then director of the Biologically Inspired Neural & Dynamical Systems lab at the University of Massachusetts in Amherst, to see if she might be interested in a collaboration. She said yes. As it happened, she had recently started thinking about the problem again, and was beginning to see how irrational-number weightings weren’t the only game in town. Anything that introduced a similar element of randomness or unpredictability might do the trick, too. “Having irrational numbers is only one way to get super-Turing power,” she says.

The route the trio chose was chaos. A chaotic system is one whose response is very sensitive to small changes in its initial conditions. Wire up an analogue neural net in the right way, and tiny gradations in its outputs can be used to create bigger changes at the inputs, which in turn feed back to cause bigger or smaller changes, and so on. In effect, the system becomes driven by an unpredictable, infinitely variable noise.

The idea is met with some skepticism. Scott Aaronson, Professor of Electrical Engineering and Computer Science at MIT, argues that models involving infinities inevitably run into trouble.

People ignore the fact that the physical system cannot implement the idea with perfect precision.

Jérémie Cabessa of the University of Lausanne, Switzerland co-authored a paper with Siegelmann published in the International Journal of Neural Systems in September 2014 which supports the idea that “the super-Turing level of computation reflects in a suitable way the capabilities of brain-like models of computation. In Brook’s article, however, he’s skeptical that such a machine is buildable.

Again, it’s not that the maths doesn’t work – it is just a moot point whether true randomness is something we can harness, or whether it even exists.

Brooks tells us that Turing often speculated about the connection between intrinsic randomness and creative intelligence.

This is not the first pairing of randomness and creativity that I’ve seen. Gregory Chaitin’s work relies heavily on randomness. Metabiology, the field he has introduced, investigates randomly evolving computer software as it relates to “randomly evolving natural software” or DNA.  And here, mathematical creativity is equated with biological creativity.  And Chaitin has remarked (probably more than once) that he doesn’t believe that continuity really works for physics theories, a perspective echoed by Aaronson.   Chaitin leans instead toward a discrete, digital, worldview.

But I find it important to take note here of the fact that the infinities of mathematics, so often problematic within physical theories, have, nonetheless, very effectively aided our imagination. The continuity of the real numbers is largely characterized by the irrational number and took years of devoted effort to be firmly established in mathematics.  In this discussion, the irrational number also opened the door to the effect of randomness in neural networks. Mathematical notions of continuity have been the mind’s way, of bridging arithmetic and geometric ideas. These bridges allow conceptual structures to develop. The roots of these ideas are in our experiences of things like space, time and object, but they somehow give the intuition more room to grow. Just a few of the fruits of their development have brought the inaccessible subatomic and intergalactic worlds within reach. Even if the world turns out not mirror this continuity, the work of Siegelmann, Redd and Younger suggests that the mind might.

Orientation through words and notation

I thought recently, again, about the relationship between the written word and mathematical notation, both being systems of marks that carry meaning. Both systems grow with usage, and both provide some steady refinement of what we are able to see. I’m not so much interested, here, in the relationship between mathematical proficiency and language proficiency, but more what cognitive processes they share that construct meaning and, more importantly, what might be their shared purposes. What can we say, for example, about the value of a good novel.

In a discussion of Michael Schmidt’s The Novel: A Biography which William Deresiewicz calls, How the Novel Made the Modern World, Deresiewicz makes the following remark:

The novel reaches in and out at once. Like no other art, not poetry or music on the one hand, not photography or movies on the other, it joins the self to the world, puts the self in the world, does the deep dive of interiority and surveils the social scope. That polarity, that tension…has proved endlessly generative.

The keys here are ‘reaching in and out at once,’ ‘joining the self to the world’ within a process endlessly ‘generative’ – generative of consistently changing views of ourselves, of the ways we organize ourselves, and of the possibilities our imagination can offer. The novel makes something new of familiar experience through the skillful arrangement of abstract, discrete and symbolic units of sound. The novel is not identified with these arrangements, but it completely relies on them.

Steven Pinker investigates the relationship between language and thought, and has described the conceptual schemes that are generalized with language. In a TED talk he gave in 2005, before the publication of The Stuff of Thought in 2007, Pinker argues that our use of language

seems to be based on a fixed set of concepts, which govern dozens of constructions and thousands of verbs — not only in English, but in all other languages — fundamental concepts such as space, time, causation and human intention, such as, what is the means and what is the ends? These are reminiscent of the kinds of categories that Immanuel Kant argued are the basic framework for human thought, and it’s interesting that our unconscious use of language seems to reflect these Kantian categories.

Shared constructions don’t care about perceptual qualities, such as color, texture, weight and speed, which virtually never differentiate the use of verbs in different constructions. This kind of generalization, Pinker says is:

a process of metaphorical abstraction that allows us to bleach these concepts of their original conceptual content — space, time and force — and apply them to new abstract domains, therefore allowing a species that evolved to deal with rocks and tools and animals, to conceptualize mathematics, physics, law and other abstract domains.

This characterization of what language does is essentially the argument in the Lakoff/Núñez book, Where Mathematics Comes From. But mathematics, like language, finds relationships within frameworks of thought that open the door to new thought. In mathematics, this happens by digging deeper into the how the mind creates any kind of structure – from the patterns in visual data, to the less understood integration of multi-modal experience.

In a 2009 publication, (Geometrein: Measuring the Environment as a Means of Orientation, in Ruedi Bauer, Andrea Gleiniger (eds.): Orientierung – Disorientierung, Lars Müller Publishing, p.282-290) Toni Kotnik makes the following observations of mathematics within a discussion of architecture and its relationship to the body orienting itself:

The limits of Euclidean geometry illustrate that the act of orientation, as an active measurement of surrounding space, determines a structure that is both geometric and metric in nature. Both structures can be understood as cognitive structures of knowledge generated by perception and experience – structures that, as individual patterns of action, shape the individual perception of the world…and, conversely, impinge on perception.

Kotnik then discusses Riemann’s famous lecture, “On the Hypotheses Which Lie at the basis of Geometry,” making the point that these ideas were strongly influenced by the work of philosopher Johan Friedrich Herbart.

According to Herbart, as people move through space, they have a variety of perceptions that do not have an immediate effect on consciousness. Herbart argues that these perceptions undergo a “graded fusion” in the mind… Riemann’s idea of manifolds renders Herbart’s psychological concepts more concrete and precise for mathematical application.

In his conclusion:

Euclid’s geometry and its universalized form in Riemann’s manifolds are examples of how mathematical concepts can be created by formalizing the act of orientation.

Here, again, mathematics is understood as the formalization of the body’s action.  Lets look at words again.  In a recent post  I discussed Peter Mendelsund’s book What We See When We Read. This takes us back to the novel.  In that post, I reproduced this of Mendelsund’s:

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.

Here the world itself is a work in progress, of synthesized discrete fragments, and reading mirrors the way we acquaint ourselves with the world.

I’ve been part of discussions among computer scientists and cognitive scientists where the question arises, to what extent is mathematics its formal representation?  Modeling mathematical relationships in both digital and analog forms suggests that the formal representation of mathematics is not, in itself, actually the mathematics. I think these are useful questions. But I will continue to argue that the meaning brought about by the discovery of mathematical structure relies on the infinite potential of notation in the same way that the meaning in story relies on the infinite potential of word constructions . Mathematics is only more abstract to the extent that in doing mathematics, one is driven to explore the depths of cognitive structures themselves, without reference to our experience. But it is the application of this exploration that has fine-tuned sensation, and brought greater depth to what we can know about our being in the world.

A mathematical philosophy – a digital view

I’ve become fascinated with Gregory Chaitin’s exploration of randomness in computing and his impulse to bring these observations to bear on physical, mathematical, and biological theories. His work inevitably addresses epistemological questions – what it means to know, to comprehend – and leads him to move (as he says in a recent paper) in the direction of “a mathematical approach to philosophical questions.” I do not have any expertise in computing (and do not assume the same about my readers) and so I am not in a position to clarify the formal content of his papers. However, the path Chaitin follows is from Leibniz to Hilbert to Gödel and Turing. With his development of algorithmic information theory, he has studied the expression of information in a program, and formalized an expression of randomness.

The paper to which I referred above, Conceptual Complexity and Algorithmic Information, is from this past June. It can be found on academia.edu. As is often the case, Chaitin begins with Leibniz:

In our modern reading of Leibniz, Sections V and VI both assert that the essence of explanation is compression.  An explanation has to be much simpler, more compact, than what it explains.

The idea of ‘compression’ has been used to talk about how the brain works to interpret a myriad of what one might call repeated sensory information, like the visual attributes of faces. Language, itself, has been described as cognitive compression. Chaitin reminds us of the Middle Ages’ search for the perfect language, that would give us a way to analyze the components of truth, and suggests that Hilbert’s program was a later version of that dream.  And while Hilbert’s program to find a complete formal system for all of mathematics failed, Turing had an idea that has provided a different grasp of the problem.  For Turing,

there are universal languages for formalizing all possible mathematical algorithms, and algorithmic information theory tells us which are the most concise, the most expressive such languages.

Compression is happening in the search for ‘the most concise.’  Chaitin then defines conceptual complexity, which is at the center of his argument.  The conceptual complexity of an object X is

…the size in bits of the most compact program for calculating X, presupposing that we have picked as our complexity standard a particular fixed, maximally compact, concise universal programming language U. This is technically known as the algorithmic information content of the object X denoted H(X)…In medieval terms, H(X) is the minimum number of yes/no decisions that God would have to make to create X.

He employs this idea, this “new intellectual toolkit,” in a brief discussion of mathematics, physics, and evolution, modeling evolution with algorithmic mutations. He also suggests an application of one of the features of algorithmic information theory, to Giulio Tononi’s integrated information theory of consciousness. As I see it, a mathematical way of thinking brings algorithmic information theory to life, which then appears to hold the keys to a clearer view of physical, biological and digital processes.

In his discussion of consciousness Chaitin suggests an important idea – that thought reaches down to molecular activity.

If the brain worked only at the neuronal level, for example by storing one bit per neuron, it would have roughly the capacity of a pen drive, far too low to account for human intelligence. But at the RNA/DNA molecular biology level, the total information capacity is quite immense.

In the life of a research mathematician it is frequently the case that one works fruitlessly on a problem for hours then wakes up the next morning with many new ideas. The intuitive mind has much, much greater information processing capacity than the rational mind. Indeed, it seems capable of exponential search.

We can connect the two levels postulated here by having a unique molecular “name” correspond to each neuron, for example to the proverbial “grand- mother cell.” In other words, we postulate that the unconscious “mirrors” the associations represented in the connections between neurons. Connections at the upper conscious level correspond at the lower unconscious level to enzymes that transform the molecular name of one neuron into the molecular name of another. In this way, a chemical soup can perform massive parallel searches through chains of associations, something that cannot be done at the conscious level.

When enough of the chemical name for a particular neuron forms and accumulates in the unconscious, that neuron is stimulated and fires, bringing the idea into the conscious mind.

And long-chain molecules can represent memories or sequences of words or ideas, i.e., thoughts.

This possibility is suggested in the light of a digital view of things. The paper concludes in this way:

We now have a new fundamental substance, information, that comes together with a digital world-view.

And – most ontological of all – perhaps with the aid of these concepts we can begin again to view the world as consisting of both mind and matter. The notion of mind that perhaps begins to emerge from these musings is mathematically quantified, which is why we declared at the start that this essay pretends to take additional steps in the direction of a mathematical form of philosophy.

The eventual goal is a more precise, quantitative analysis of the concept of “mind.” Can one measure the power of a mind like one measures the power of a computer?

Quantification as a goal can be misunderstood. To many it signifies a deterministic, controllable world. Chaitin’s idea of quantification is motivated by the exact opposite. His systems are necessarily open-ended and creative. Quantification is more the evidence of comprehension.

There is one more thing in this paper that I enjoyed reading.  It comes up when he introduces the brain to his discussion of complexity.  I’ll just reproduce it here without comment.

Later in this essay, we shall attempt to analyze human intelligence and the brain. That’s also connected with complexity, because the human brain is the most complicated thing there is in biology. Indeed, our brain is presumably the goal of biological evolution, at least for those who believe that evolution has a goal. Not according to Darwin! For others, however, evolution is matter’s way of creating mind.   (emphasis added)

 

Architecture, orientation and mathematics

Recently, I became intrigued with the discussions of topology that I found among architects and historians of architecture. I saw a few familiar threads running through these discussions – like the emergence and self-organizing principles of biology, together with the view that mathematics was not, primarily, a tool but more a point of view.

I was introduced to the term bioconstructivism in a 2012 paper by John Shannon Hendrix: “Topological Theory in Bioconstructivism.”

Bioconstructivism involves the engagement in architecture of generative models from nature. This is in the tradition of natura naturans in architecture, which is the imitation of the forming principles of nature, as opposed to natura naturata, the direct imitation or mimesis of the forms. According to Plotinus in the Enneads, it is the purpose of all the arts to not just present a “bare reproduction of the thing seen,” the natura naturata, but to “go back to the Ideas from which Nature itself derives”

…An important element of Bioconstructivism is autopoiesis or self-generation, taking advantage of digital modeling and computer programs to imitate the capacity for organisms in nature to organize themselves, or for unorganized or fluid material to consolidate itself, based on the inner active principle of the organism, an “essential force” or “formative drive” which contradicts the mechanistic theories of Galileo, Descartes and Newton. The monad of Leibniz, for example, can self-generate in “integrals” from pre-existing sets of variables, resulting in “continuous multiplicity.”

But I must admit that, from this paper, I didn’t get a clear sense for how topology was used or understood in either architectural design or critique. However, in a 2007 essay on this very topic (on the website RZ-A), I found the following:

Toni Kotnik (the author of ‘The Topology of Type’) believes that the only goal for introducing topology to architecture has been ‘to overcome dialectical strategies of homogeneity or heterogeneity, which dominated the architectural discussion throughout the 20th century’ but rather than giving a real answer to these debates ‘a superficial practice has been established in which every non-linear deformation of the usually used canon of forms gets classified as topological design both by the architect and by the critics.’ (17) He also thinks that the current architecture is not really based on topology but ‘on differentiable dynamical systems and popular spin-offs like chaos theory and fractal geometry’ and suggests that ‘a topological approach to architecture should not be seen as a form-generating tool but as an abstract form of thinking to structure sensorial and rational perceptions in a spatial way’.

I searched for “The Topology of Type,” but found instead, Kotnik’s piece “…there is geometry in architecture.” And here, I was excited to see, a provocative discussion of mathematics and architecture, broader and more interesting than the more specific discussions of various applications of topological ideas.

Kotnik begins with this:

Within contemporary architectural design the form – rule relationship is often understood as the application of geometric rules in a generative process of form-finding, that is rules are a logico-algebraic text out of which architectural form emerges through the manipulation of data. By looking at the etymological roots of mathematics another reading of geometry can be uncovered that relates geometry back to bodily experience and the question of spatial orientation. This enables the re-introduction of the body into contemporary discourse of digital architecture.

Kotnik makes the point that, historically, geometry has been viewed as something that architects use (or consume), not something that they produce. The introduction of digital computing in architecture, however, opens the door to “the emergence of architectural form out of the manipulation of data” which, Kotnik observes, could introduce scientific thinking and methodology into the design process. He moves, then, into the discussion of mathematics that I so much enjoyed, beginning with the etymology of the word mathematics.

“…mathematics has its roots in the Greek ta mathemata, which means what can be learned where learning, mathesis, is about the recognition of the unchanged, the stable, of the Being in a world of constant Becoming.” Kotnik argues that mathematic is “the human search for patterns as a means of orientation,” and so “in its original meaning, mathematics is about relating the body with the world around. As such, mathemata is about orientation.”   (emphasis added)

Kotnik briefly surveys the developments in mathematics within cultural histories. Euclidean geometry is seen as “an act of physical orientation based on the creation of an intellectual structure,” shaping the individual perception of the world. He calls attention to Riemann’s reformulation of the foundation of geometry, his concept of manifold, and the influence of philosopher Johann Friedrich Herbart is noted. Herbart understood that the variety of perceptions people have, as they move through space, undergo a “graded fusion,” that “glues” individual perceptions together to form a geometric image. Related to the argument I made in Cognition, brains and Riemann, Kotnik suggests

Riemann’s idea of manifolds renders Herbart’s psychological concepts more concrete and precise for mathematical application.

…Euclid’s geometry and its universalized form in Riemann’s manifolds are examples of how mathematical concepts can be created by formalizing the act of orientation.

This discussion of mathematics and architecture rests on the idea that mathematics is not primarily a tool, but almost an articulation of sensation and its consequences. Mathematics, perhaps, can be seen as providing alternative ways to organize sensation or, more to the point, as alternative ways to understand what we see.

I’ll conclude with this last bit from Kotnik:

Measuring space is a central activity for structuring our surroundings and an important orientation element in the human environment. However, this environment must not be understood in a limited way as the purely physical environment. It must be seen more broadly and comprehensively as a multilayered perceptual and experiential space. This expanded understanding of space and the incorporation of the subject into space transforms measurement into an act of individual orientation. The importance of such bodily measuring of space,of geometrein, is not only justified by philosophers like Heidegger, Merleau-Ponty or Deleuze but also by developments in contemporary neuroscience. What emerges is an understanding of architecture as a primarily emotional and perceptual experience grounded in biological values that have to be put forward in the design process.

The mathematical nature of self-locating

A 2011 TED talk in London was brought to my attention recently. The speaker, Neil Burgess  from University College London, spoke on the topic, “How your brain tells you where you are.” Burgess investigates the role of the hippocampus in spatial navigation and episodic memory. In the talk he describes the function of what are called place cells, boundary-detection cells, and grid cells. I wrote (also in 2011) about a Science Daily report on studies dedicated to understanding how hippocampal neurons represent the temporal organization of experience, in particular, how they bridge the gaps between events that are not continuous. But here I want to focus more on how the brain constructs spatial experience.

Electrophysiological investigations have identified neurons that encode the spatial location and orientation of an animal. Among these are those that have come to be called place cells, head direction cells, and grid cells. Burgess also talked about boundary detection cells in his 2011 TED talk, which seem to be coordinated with head direction. I was particularly struck by the clarity of some of the data images he presented. He showed his audience images produced by the firing of boundary detection cells in response to boundaries in the environment of a rat. In one of them we could see cell firings in response to one of the walls in the rat’s environment. In another, created after a second wall had been added to the environment, the firing was duplicated with respect the added wall. Burgess also presented the image of cells that fired when the rat was about midway between the walls, and one could see directly that when the rat’s box was expanded, the firing locations expanded. These boundaries needn’t be rectangular walls. They can be the drop at the edge of a table or the circular wall of a circular box.

Grid cells are creating representations a little differently, in a quasi-mathematical way, as their name suggests. Burgess tells about the rats again:

Now grid cells are found, again, on the inputs to the hippocampus, and they’re a bit like place cells. But now as the rat explores around, each individual cell fires in a whole array of different locations which are laid out across the environment in an amazingly regular triangular grid…So together, it’s as if the rat can put a virtual grid of firing locations across its environment — a bit like the latitude and longitude lines that you’d find on a map, but using triangles. And as it moves around, the electrical activity can pass from one of these cells to the next cell to keep track of where it is, so that it can use its own movements to know where it is in its environment.

Both boundary detection cells and grid cells reflect a sensory perception of the environment. But neurons also encode movement from proprioceptive information that can be used to measure the body’s displacement as we move (path integration). This is not movement defined by the environmental changes that occur, but from the body’s sensations of itself.
In a more recent paper Burgess and co-author C. Burgess  describe an interesting test of the errors that can be produced by the iterative neural processing of self-motion.

This process, known as path integration or dead reckoning, requires the animal to update its representation of self-location based on the cumulative estimate of the distance and direction it has traveled. It can be shown that an animal is utilising path integration by introducing a known error into its representation of direction or distance: in the case of the gerbils, if they are rotated prior to the return leg of the journey, and this is done slowly so that the vestibular system does not detect the motion, then the animals head towards the nest with an angular error equal to the amount they were rotated by.

So when we try to find our way back to something, like where we parked the car, we likely use boundary-detecting cells to remember distances and directions to buildings and boundaries, but we also remember the path we took, represented by the firing of grid cells and path integration. The interaction of these things seem to contribute to the pattern of neural firings that become associated with a particular place, a cognitive map of that place, formed by what are called place cells. There is a nice discussion of the history of the study of place cells which includes a number of images at BrainFacts.org. There the point is made that the ‘cognitive map’ defined by place cells is a ‘relation among neurons,’ not among points in space.

In brief, we can think of the “map” of a session in terms of space (the spatial relations of firing fields) and time (the tendency for pairs of cells to fire together or not). Since the speed of rats is restricted, these are essentially equivalent. An important concept is that the map is entirely in the brain. In this description, a map is defined by the relation among hippocampal neurons, not by the relationships between neurons and the environment. The linkage to the environment is critical, but does not define the map.
The temporal relations are important for two reasons. First, neurons in the brain do not know about space directly, but they know about time. Neurons can code the timing relations of the neurons that project to it, but not the spatial relations. In other words, within the brain, the map is a timing map that encodes the temporal overlap between cell pairs.

 

There are a few interesting things going on here. No doubt the grid cell idea and the vector-like measures of displacement that are encoded when we move around, trigger memories of mathematics. They are like our mathematical analyses of the 3-dimensional space of our experience. Place cells, on the other hand, are like another level of abstraction. They seem to have more in common with coding and non-spatial analyses, even though we don’t seem to know how they do what they do. The neurons that fire to represent a particular location have no spatial relationships among themselves. Neighboring place cells do not indicate neighboring environmental areas. And while correlated to sensory input, they are part of a non-sensory system. It is the integration of this system with the neural representations of boundaries, direction, and distance (among other things) that create our spacial awareness. Certainly sensory information is being subject to some kind of transformation. Spatial relations are translated into what look like purely temporal ones (the timing of neuron firing). The non-sensory system then stores a coded representation of a sensory one. Here again we see, not the mathematical modeling of brain processes but more their mathematical nature.

What we see when….

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.

 

What mathematics can make of our intuition

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.

Embodiment and a Philosophy of Mathematics

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.

 

Mathematics in the light of Maturana’s biology of cognition

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.

And also:

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.

The mathematics of common sense

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.