Neuroscience and Riemann

I would like to go back today to Riemann, and the significance of his generalized notions of space and magnitude, but with an eye on what neuroscience may be adding to how mathematics gains its effectiveness.

In a recent post, I pointed to the influence the philosopher Herbart had on Riemann’s 1854 lecture in which Riemann proposed the most generalized notions of magnitude and space. I pulled the following excerpt from a book by David Cahan on the foundations of nineteenth century science:

Herbart argues that each modality of sense is capable of a spatial representation.  Color could be represented as a triangle in terms of three primary colors, tone as a continuous line, the sense of touch as a manifold defined by muscle contractions and still other spaces as associations of hand-eye movements.  “To be exact,” he wrote, “sensory space is not originally a single space.  Rather, the eyes and the sense of touch independently from one another initiate the production of space; afterward both are melted together and further developed.  We cannot warn often enough against the prejudice that there exists only one space, namely, phenomenal space.”  Therefore, for Herbart, “space is the symbol of the possible community of things standing in a causal relationship.”  He insisted that for empirical psychology space is not something real, a single container in which things are placed.  Rather, it is a tool for representing the various modes of interaction with the world through our senses.

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.

I did a little looking into the brain circuitry being talked about and found that a neuronal analysis of the relatedness of number and space in our experience strengthens Herbart’s perspective and could suggest something about the nature of Riemann’s insight.  I mean to be careful here. I don’t want to be sloppy about my references to either the mathematics or the neuroscience, but I find it worth pointing to some interesting correspondences.

Quite a lot of work has been done on the spatial aspect of number and some of this is described in the 2005 article.   In a 2009 article, Domenica Bueti and Vincent Walsh explore the parietal cortex and the representation of time, space, number and other magnitudes and suggest the following:

Imagine you are Darwin for a day and you are charged with granting a species the ability to count. Where would be the most efficient place in the brain for this discrete numerical system? The parietal cortex is already equipped with an analogue system for action that computes ‘more than–less than’, ‘faster–slower’,‘nearer–farther’, ‘bigger–smaller’, and it is on these abilities that discrete numerical abilities hitched an evolutionary ride.

The work of the parietal cortex is discussed in a 1995 issue of The Neuroscientist (by Charles Gross and Michael Graziano of Princeton).  A number of things come together in the posterior parietal cortex:

In summary, the posterior parietal cortex is where vision, touch, and proprioception come together for the first time.  It is the hub of a system for the processing of spatial information.  This system includes not only several regions within the parietal cortex….but a widespread network of other cortical and subcortical areas, including the ventral premotor cortex, the putamen, the frontal eye fields, the superior colliculus, the hippocampus, and principal sulcus.  These areas are specialized for a variety of different spatial functions, such as visuomotor guidance of limb, eye, and head movements, navigatinig in the external environment and holding recent memory about the location of objects in space.  They appear to carry on, in specialized fashions, the processing of information about space that is begun in the parietal cortex.

What I find most striking in reading the literature is the extent to which the body creates structure out of  multi-sensory interaction.  And what we call magnitude, space and number emerge from these multi-sensory interactions.  This is not, I think, dissimilar from Herbart’s insight.   Brain circuitry handles the body’s need for orientation, the perception of distance, speed, size and quantity interactively, with both internal and external sensory systems.  Much of this work means to understand the evolution of the conceptual side of our nature, our idea-driven experience.  One of the conclusions of the Bueti/Walsh article is this one:

…it was suggested that different magnitudes originated from a single developmental algorithm for more than–less than distinctions of any kind of stuff in the external world. The development of magnitude processing proceeds by interactions with the environment and is therefore closely linked with the motor reaching, grasping and manipulating of objects. It was further suggested that the emergence of our ability to manipulate discrete quantities evolved from our abilities with continuous quantities.

Riemann was not engaged in a psychological analysis of the evolution of concepts, but he was interested in the essence of ideas which appeared to be experience driven.  And the greatest generality would be found in the rendering of their most fundamental attributes. I would like to suggest that his careful analysis of what one could mean by magnitude or what one could mean by space holds within it remarkable insights into interactive chains of cognition itself.  But these can be found in the purely abstract investigation of mathematical worlds.

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