Mathematical hybrids and the like

My attention was just recently brought to the work of philosopher and poet Emily Grosholz.  It’s rare to find an individual so steeped in the ways of poetry and mathematics, and the desire to explore how and what they express about us.  What I would like to consider here, in this particular post, is really a detail of the extensive thought and research that Grosholz brings to the discussion of how mathematics grows.  But I think it’s a powerful idea that can have a good deal to say about how we work, and how we, as a species, produce the bountiful and variegated products of human culture.

Grosholz is the author of many books that include works on the philosophy of mathematics as well as works of poetry. Her latest is Starry Reckoning: Reference and Analysis in Mathematics and Cosmology.  What follows is based on a piece that she contributed to a book she edited with Herbert Breger. The book is called The Growth of Mathematical Knowledge, and her piece is given the title The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge.  Here Grosholz argues that unlike what has been considered before, different branches of mathematics do not reduce to other branches.  Philosophers of mathematics have discussed the possibility that geometry can be reduced to arithmetic, arithmetic to predicate logic, and arithmetic and geometry to set theory. This is understood in much the same way that one might claim that biology can be reduced to chemistry and chemistry to physics.  The vocabulary of the reduced theory is redefined in terms of the reducing theory.  In the sciences, the reducing theory has been thought to play an explanatory role, suggesting an inherent unity among the various scientific disciplines.  But in mathematics the so-called reducing theory is not used so much as an explanation of the reduced ideas, but more as a foundation for them.  And mathematicians have long had difficulty with foundational questions.   Grosholz, on the other hand, proposes that mathematics is a collection of rationally related but autonomous domains and then highlights the potent role of what she calls mathematical hybrids.

She explains that in Greek mathematics the autonomy of domains is clear.  Geometry is about points, lines, planes, and figures, and geometric problems involve relations between parts of the whole of spatial figures.  Arithmetic is about numbers, and problems in arithmetic involve monotonic, discreet succession.  The vocabulary of logic is one of terms, propositions, and arguments, and problems in logic involve ideas of inclusion, exclusion, consistency, and inconsistency.   While these separate domains may seem to resist assimilation, 17th century mathematics introduced some unifications.  Among these unifications is Descartes’ application of algebraic techniques to geometric constructions, and Leibniz’s application of combinatorics to an analysis of curves.  Grosholz spends some time on each of these.  She points out that Leibniz was fascinated with formal languages and number theory, and that he believed that the art of combinations was central to the art of discovery.  She argues that Leibniz’s investigation of algebraic forms in the calculus is grounded in “an imperfect but suggestive analogy between numbers and figures.”  The infinite summing of infinitesimal differences, that becomes the integral, emerges from his ability to bridge geometric ideas about a curve (like tangent, arc length, area), with algebraic equations, and through the notion of an infinite-sided polygon approximating the curve, patterns of integers were also connected.  Here the mathematical hybrid emerges: an abstract structure that rationally relates different domains in the service of problem solving.  On a deeper level, objects in each domain must actually exhibit features of both domains, despite the instability created by their differences. But, Grosholz argues, this instability does not mean that hybrids are defective.  They are held together by the clarity of the domains from which they emerge, and the abstract structures that link them.  “Logical gaps are to be found at the heart of many hybrids,” Grosholz explains, but imaginative analogies inspire the kind of revision and invention that promotes the growth of mathematical knowledge.

I was always impressed by the fact that these intuitive leaps that Leibniz took, while prompting subsequent generations to feel the need to bring acceptable rigor to the notions, were nonetheless substantiated.  Grosholz lends some important detail to the picture Richard Courant paints of 17th century pioneers of mathematics in his classic text, What is Mathematics?

In a veritable orgy of intuitive guesswork, of cogent reasoning interwoven with nonsensical mysticism, with a blind confidence in the superhuman power of formal procedure, they conquered a mathematical world of immense riches.

This talk of hybrids reminded me of the interdisciplinarity that Virginia Chaitin writes about.  I wrote this in an earlier post about one of her papers:

What she proposes is not the kind of interdisciplinary work that we’re accustomed to, where the results of different research efforts are shared or where studies are designed with more than one kind of question in mind. The kind of interdisciplinary work that Chaitin is describing, involves adopting a new conceptual framework, borrowing the very way that understanding is defined within a particular discipline, as well as the way it is explored and the way it is expressed. The results, as she says, are the “migrations of entire conceptual neighborhoods that create a new vocabulary.”

In her own words:

…an epistemically fertile interdisciplinary area of study is one in which the original frameworks, research methods and epistemic goals of individual disciplines are combined and recreated yielding novel and unexpected prospects for knowledge and understanding. This is where interdisciplinary research really proves its worth.

Grosholz’s identification of the hybrid is an important insight, and I would argue that it has implications beyond mathematics.  It may be that because the objects of mathematics are so clean, or unambiguous, the value of the hybrid is more easily observed.   But my hunch is that productive analogies likely belong to the stuff of life itself.

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