Gauss, Riemann and Einstein: Neurons Reaching Behind Experience

I had the opportunity to listen to Paul Churchland when he gave a talk last Friday, on Cognitive Enhancement, at the University of Texas at Dallas.  He used the time to address, not enhancement drugs or exercises, but the enhancement effects of language and symbol.  I poked around today to find more more on the ideas that he only had time to reference and I stumbled upon Dudley Shapere’s critique of Churchland’s book The Engine of Reason, the Seat of the Soul.

I was most interested in Churchland’s conviction that cognition is not based in “sentence crunching” or logic-like processes, but is grounded more in small neuronal pattern adjustments of synaptic connections. This is consistent with my sense that mathematics is born more of pattern and image than logic.  But language as the externalization of learning, makes the transference of cognitive acquisitions to another individual possible and also presents explorers with new ground to ponder.  In his book, Churchland asks some interesting questions:

How is it that human cognition manages to reach behind appearances?  How do we discover, for example, that light consists of submicroscopic waves?…..How then does a neural network such as a human scientist – a network doomed, after all, to be trained on a uniform diet of observable phenomena – ever manage to form concepts or prototypes of unobservable phenomena?

The answer, to a first approximation is that we learn all of our prototypes solely within the domain of observable things.

Shapere argues strongly that recycling prototypes is not enough to account for how the world gets opened up by the sciences.  To describe some of his difficulty with this notion, he uses the idea of curvature in mathematics, of which, he says, the commonsense view is in terms of space.  In other words, the curvature of a one-dimensional line is generally imagined by seeing it on a two-dimensional surface, and the curvature of a two dimensional surface is imagined by seeing it in a three dimensional space. He continues:

In this commonsense view, it makes no sense to speak of the curvature of our three-dimensional space, because we cannot access a four-dimensional containing space with respect to which we could say it curved.

He goes on to say:

However, Gauss’s theory of surfaces, generalized by
Riemann to spaces of any dimensionality, introduced a notion of curvature capturing the important features of the commonsense view except for being intrinsic to the space (line, surface, etc.), determinable by measurements
entirely within the space itself, without appeal to anything about any higher-dimensional containing space, even its existence.

This is the idea Einstein would use.  “Such transformations,” Shapere argues, “are uncapturable in today’s neural network models.”

I want to agree, although I, myself, am captivated by the relationship between the original, spatial notion of curvature and the later, very powerful, transformation of it.  And I’m convinced that neuroscience will make a significant contribution to how we understand what we do, particularly when it reveals unexpected relationships, like the fact I learned today that the visual cortex is recruited by blind individuals when they learn to read Braille.  In these individuals, the visual cortex is processing information derived from touch.  We could say the brain is being efficient since the visual cortex in these individuals is not being used.  But I think it’s also important to note that Braille is the touch version of the visual word.

The symbol is most important to Shapere’s understanding of scientific investigation.

Neither the character nor the methods of science can be understood in terms of individual thought-processes, with their attendant prejudices, myths, and misunderstandings; they must be understood in terms of an objective body of
doctrines and evidence developing sometimes in radical directions. Herein lies the importance of language, logic and, I would add, mathematics, which gives us our only means of conceptualizing that which lies well beyond our middlesized

It’s not the individual, he says,

For a primary attribute of the modern scientific
knowledge-seeking process, and of the knowledge that results from it, is its independence of the individual.

I’ll say it again, because I like it so much, from last week’s post:

The symbol jumps over the blind spot of cognition with a flash of unexpected light.

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