Infinities, metaphors and being human

Our thoughtful, imaginative worlds are married to our physical experiences but the subtleties of their union are almost impossible to fully appreciate.  Mathematics, I often argue, has the potential to provide a better view of the situation, perhaps because of the inexhaustible depth of its abstraction, together with the precision it brings to a concept, and some of the surprising fruits of its application.   A recent paper from Roy Wagner at The Hebrew University of Jerusalem comes at it, again, from a more formal, philosophical perspective rather than a cognitive science perspective.   The paper, Infinity Metaphors, Idealism, and the Applicability of Mathematics, highlights the plurality of conceptual frameworks within mathematics, particularly with respect to infinities, and uses this to make an interesting point.  Wagner quickly addresses the book Where Mathematics Comes From from George Lakoff and Rafael Nunez and surveys the short-comings in how they account for the notion infinity.  He produces mathematical notions of infinity that can’t be captured by the Basic Metaphor of Infinity (BMI), proposed by Lakoff and Nunez in their naturalistic account of mathematics. This BMI is the cognitive mechanism underpinning our concept of infinity in mathematics.  The idea is that the concept of continuous processes is a broadening of our experience of iterative processes, processes that go on and on. Among other examples, Wagner points out that the uncountable infinity of real numbers is not captured by the BMI.

Uncountable, strongly inaccessible cardinals are defined in such a way that they cannot be constructed from smaller cardinals, namely, they cannot be the end state of any infinite process constructable from Aleph0. In a way, they are conceived, precisely, as that which cannot be conceived through BMI! The metaphor here is not that of a final state of an indefinite process, but of that which is beyond any final state of any indefinite process, including the indefinite process of constructing ever larger infinities – a figure of transcendence that’s not captured by BMI.

Wagner also points out that, while Lakoff and Nunez apply this iterative process idea to the point at infinity in projective geometry, it doesn’t actually work.

To make one final point, let’s consider one more example. Henderson (2002) observes that explaining the projective point at infinity in terms of BMI (as Lakoff and Núñez do) would yield two intersection points at infinity for any pair of parallel lines (one point in each direction). So BMI cannot be the origin of the single projective point at infinity. Indeed, the origin of this point at infinity is easy to track down: it comes from the point of perspective in Renaissance art and draftsmanship – a practical tool, rather than an inference carried between domains. While expressing a commitment to the notion of embodied cognition, Lakoff and Núñez’ notion of metaphor largely ignores the origin of mathematical ideas in culturally constrained practices with tools.”

(The Henderson reference is to a review of the Lakoff/Nunez book that appeared in the Mathematical Intelligencer in 2002).

I was happy to see Wagner point to the investigation of perspective in Renaissance art as the origin of the point at infinity.  In this light, the idea looks like a tool rather than a metaphor, and this will support the thesis Wagner later argues.  But even this doesn’t do justice to what captivates me about this development in mathematics.   The point at infinity completes the mathematical idea of parallel vs. not parallel by examining an infinity that can’t actually be reached. (which had been one of the difficulties with Euclid’s parallel postulate). At the same time, the source of inspiration for this examination came from th visual experience of that same receding distance, from how it appears to the eye.

A large part of Wagner’s paper is a discussion of the work of 19th century Polish philosopher and mathematician Jozef Maria Hoene Wronski.  When Wagner begins to sum things up he says this:


Wronski’s form of idealism (like other forms of German idealism, but with a mathematical edge) thinks of being not as given, but as formed by the knower, who is in turn formed by that being. Of course, neither being nor knowledge are formed arbitrarily, but being is carved out with the tools and concepts of the knower…Tools and concepts enable humans to inhabit the world in different ways and carve out different observations and forms of understanding. It is in this sense that reason can constitute and impose on being, rather than content itself with a regulative role. As knowledge evolves, so do our tools, ways of intuiting, and the phenomena we encounter and create.

While this is a philosophical piece, it certainly brings to mind the perspective in biology that was pioneered by Humberto Maturana and Francisco Varela.  In their book, The Tree of Knowledge. they describe cognition as “an ongoing bringing forth of a world through the process of living itself.” But in Wagner’s paper this idea has an interesting consequence.

This approach allows us to extend the plurality inherent in mathematical language to what can be conceived as a plurality inherent in the ways humans form being. Instead of a single “mathematics” miraculously corresponding to a single “physics,” we have an inherently multiple mathematical language that corresponds to some way of inhabiting our universe.

This suggests that there are ways of inhabiting our universe that we do not yet see, but may already have been given shape in mathematics, universe in our imagination.

Describing mathematical knowledge as part of the physicists’ instruments span, Wagner characterizes physics in a way that Galileo could not have forseen.  He proposes that

Physical phenomena are not simply observed by physicists, but also constituted by the possibilities that the physicists’ instruments span…In a way, modern physics is inherently designed to discover precisely that portion of our way of inhabiting the universe that can be discovered through mathematical analogies.

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