I spy the confluence of mathematics, psychology, and physics

I find the relationship between mathematics and vision fascinating.   Even within mathematics itself, seeing how the geometric expression of ideas can clarify or further develop countless mathematical thoughts is always worth noting – like the graphs of functions, or the projections of figures.  I’ve written before about the relationship between the brain’s visual processes and mathematics.  And, along these lines, I had reason to look a little further into Hermann von Helmholtz’s contributions to both vision and mathematics.

Nineteenth century France and Germany broke from past ideologies, and new economic and political structures emerged.  There were significant developments in science and mathematics and significant growth in specializations.  I’ve highlighted the work of Bernhard Riemann often, paying particular attention to his famous 1854 paper On the hypotheses which lie at the bases of geometry, to some extent because Riemann acknowledged the influence of philosopher Johann Frederich Herbart who pioneered early studies of perception and learning.   I wrote a piece for Plus Magazine that suggested parallels between Riemann’s insights into the nature of space, quantity and measure in mathematics, and modern studies in cognitive neuroscience that address how number, space, and time are more the result of brain circuitry than features of the world around us.

It became particularly clear to me today that another nineteenth century heavyweight, whose multidisciplinary research spans physics, psychology, and mathematics, was similarly influenced by Herbart.  In an essay with the title The Eye as Mathematician, Timothy Lenoir discusses Hermann von Helmholtz’s mid-nineteenth century theory of vision, which suggests an intimate link between vision and mathematics. And Lenoir explains that Helmholtz’s theory of vision was “deeply inspired by Herbart’s conception of the symbolic character of space.”

Lenoir sketches out how the brain uses the data it receives to construct an efficient map of the external world. The data may include sensory impressions of color, or contour along with, perhaps, the registration of a light source on a peripheral spot on the retina.  The location of the light source is then defined by the successive feelings associated with eye movements that bring the focal part of the eye in line with the light. A correspondence between the arc defined by each positional change in the eyes, and the stimulation of that spot on the retina, is stored in memory and repeated whenever that spot on the retina is stimulated.  Helmholtz called these memories local signs.  They are learned associations among various kinds of sensory data that also include head and body positions. From sequences of sensory inputs, the mind creates pictures, or symbolic representations, that provide a practical way for us to handle the world of objects we find around us. Helmholtz is clear, however, that these pictures or symbols are not copies of the things they represent. While causally related to the world around us, the quality of any sensation belongs to the nervous system. For Helmholtz, the things we see are a symbolic shorthand for aggregates of sensory data like size, shape, color, contrast, that have become associated through trial, error and repetition. The more frequently associations occur, the more rapidly linkages are carried out. Symbols then become associated with complexes of sensory data. And, like a mathematician, the brain learns to operate with the symbols instead of with the production of the complex of sensory data directly. This, Helmholtz argued, is how the constructive nature of perception becomes hidden and nature seems to us to be immediately apparent.

There are other psychological acts of judgment in Helmholtz’s visual theories. The brain has to decided whether a collection of points, for example, generated by stimulation of the retina, does or does not represent a stable object in our presence. To be an object, the points registered on the retina would need to be steady, to not move or change over time. The brain tests their stability by evaluating successive gazes or passes over the object. According to Helmholtz, the collection of points is judged to be a stable object if the squares of the errors, after many passes, are at an acceptable minimum. This is meant in the same sense as the principle of least squares in mathematics. Lenoir calls these measuring mechanisms sensory knowledge, “part of the system of local signs we carry around with us at all times…”

Lenoir’s piece also made it clear that, in the mid 1800’s, there was significant overlap in the methods and the instruments developed by physiologists and astronomers.  Gauss introduced the use of least squares in astronomy.  Helmholtz invented the ophthalmometer, an instrument that measures how the eye accommodates changing optical circumstances, which makes the prescription of eyeglasses possible.    He described the ophthalmometer as a telescope modified for observations at short distances.

In an article for the Stanford Encyclopedia of Philosophy, Lydia Patton also addresses Helmhotz’s work in mathematics.

Even when he was writing about physiology, Helmholtz’s vocation as a mathematical physicist was apparent. Helmholtz used mathematical reasoning to support his arguments for the sign theory, rather than exclusively philosophical or empirical evidence. Throughout his career, Helmholtz’s work is marked by two preoccupations: concrete examples and mathematical reasoning. Helmholtz’s early work in physiology of perception gave him concrete examples of how humans perceive spatial relations between objects. These examples would prove useful to illustrate the relationship between metric geometry and the spatial relations between objects of perception. Later, Helmholtz used his experience with the concrete science of human perception to pose a problem for the Riemannian approach to geometry.

As I read, I felt like I was enjoying just a little sip of the rich confluence of physics, psychology and mathematics. We keep trying to unravel the tight weave that binds the nature of the world, the nature of our perception and experience, and how we pull it all together.

2 comments to I spy the confluence of mathematics, psychology, and physics

  • Another good scout to consult at this confluence is the polymath Charles S. Peirce (1839–1914). Just off hand, here’s a link to a series of posts I wrote on the connection between Riemann’s concept of a manifold and Peirce’s theory of triadic sign relations.

    Sign Relational Manifolds

  • A bit tangential, but this all reminds me of a fascination I used to have with the linkage between math and the prosody, rhythm, and spacing of language… i.e. spoken language is a quite rapid-fire continuous stream of sounds that our brains effortlessly break into meaningful units, because of intonation, stress, patterns, gaps/pauses etc. in that stream — there must be some mathematical rules involved in those patterns that, remarkably, we learn/internalize at a very young age for our native tongue, and apply unconsciously, but I think little is understood of it. Helmholtz’s “…the constructive nature of perception becomes hidden and nature seems to us to be immediately apparent” applies to speech. Incredible how we all learn to speak and comprehend so early in life, yet with virtually no awareness of how we do it!