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Visualizing, Metaphors and Mathematics

I’ve thought that one of the reasons it’s difficult to resolve questions about the nature of mathematical reality is that we’re not exactly clear on what it means to ‘perceive’ something.  Trying to establish whether or not even the data of our senses is somehow independently ‘real,’ has fueled centuries of philosophical debate.  I found a paper today with the title Visualism in Mathematics by Fernando Flores of Lund University.  The paper applies the visualism of Professor Don Ihde to mathematics (from his book Expanding Hermeneutics, Visualism in Science).  It’s an interesting read in that it also addresses how the formalism of modern mathematics is, in itself, visual and the rigor of logic based conceptualizations just hides its “direct connection to the intuition of everyday life.”   One of the passages I liked was this one about Dedekind’s cuts, (used to give a firm logical foundation for the real number system):

There is a very important and unconscious manipulation of text-depicting gestalts in Dedekind‘s construction, the praxis of cutting and separating, the praxis of finding things spread around in suitable successions make this proof a master piece of art rather than a scientific result. That talks a lot about the nature of mathematical knowledge, which is in fact deeply rooted in the everyday world.

Flores also suggests a sameness between mental and physical space and early in the paper says the following:

A common denominator of all these visual constructions is to represent a certain type of ‘logic visual reality’, which could be illustrated by John Venn‘s (1834 –1923) configurations of circles. The geometric constructions in logic, works generally as analogies but they are more than that, they are parallel phenomenological worlds.

The idea that abstract constructions are objects in a perceived world makes the old questions, like what’s really there and what did we invent, harder to separate.

For me, these kinds of thoughts inevitably connect to the work of cognitive scientists like Mark Turner and Gilles Fauconnier who look closely at the details of how conceptual worlds are built with metaphor-based mental processes.  Some of this can be found in their paper Rethinking Metaphor.

But the conceptual is pushed right back into nature in work known as biosemiotic mimesis, where metaphor is, essentially, the way the organism perceives.  This view of cognition is discussed in a paper by Andreas Weber from Hamburg, Germany called Mimesis and Metaphor.  This work has its own history (some of which you can see in the paper’s references).  I’ll save this discussion for another post, but I will take from this paper what is consistent with the theme of this one, that the power of this thing we call metaphor is that it also brings us something new, something that wasn’t there before and can now be perceived.  By Weber it is described in this way:

The deciding moment in the symbolic (or poetic) achievement, the so called “keen metaphor” (Haverkamp 1995), is that it does not only arouse a vital import through the synaesthetic enactement of feeling: it produces something entirely new, something never heard of, which becomes an opaque part of the world itself.

And also:

The symbol jumps over the blind spot of cognition with a flash of unexpected light. It knows more than what it was borrowed for. Its wisdom stems from a double source, always merging body and culture.

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