There are countless ways to explore what may be called the two faces of mathematics – algebra and geometry. Modern mathematical systems have their roots in both algebraic and geometric thinking. Like the organs of the body which are built on the redirected sameness of cells, algebra and geometry live in all manner of relationship in modern mathematics. But I found a particularly novel focus on their distinctness in an article by Peter Galison (published in an Autumn 2000 issue of Representations by the University of California Press).
The article looks at the intriguing conflict between two things – that Paul Dirac considered himself a geometer and that there are no visual, let alone geometric, presentations in his publications or lectures. Galison says of Dirac:
Lecturing in Varenna, also in the early 1970s, he recalled the “profound influence” that the power and beauty of projective geometry had on him. It gave results “apparently by magic; theorems in Euclidean geometry which you have been worrying about for a long time drop out by the simplest possible means” under its sway. Relativistic transformations of mathematical quantities suddenly became easy using this geometrical reformulation. “My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e.g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form.”
The article considers the effect that social and psychological choices can have on science and mathematics and, in this case, the suppression of projective geometry. Galison says of his idea:
My inclination, then, is to use the biographical-psychological story not as an end in itself, but rather as a registration of Dirac’s arc from Bristol to Cambridge, to an identification with Bohr’s and Heisenberg’s Continental physics. In that trajectory, Dirac was sequentially immersed in a series of territories in which particular strategies of demonstration were valued.
There are a few things I find interesting about this historical look at emerging thoughts. One is the way mathematics can be seen as the way to greater things, illustrated by these remarks:
Projective geometry came to stand at that particular place where engineering and reason crossed paths, and so provided a perfect site for pedagogy.
or
In 1825, Dupin proclaimed in his textbook that geometry “is to develop, in industrials of all classes, and even in simple workers, the most precious faculties of intelligence, comparison, memory, reflection, judgment, and imagination…. It is to render their conduct more moral while impressing upon their minds the habits of reason and order that are the surest foundations of public peace and general happiness.”
or this remark from De Morgan in 1868:
“Geometry is intended, in education,. . . to [unmask] the tricks which reason plays on all but the cautious, plus the dangers arising out of caution itself.”
Another is how judgments of correctness are made within mathematics community, referenced with this observation:
Geometry did not, however, survive with the elevated status it had held in France at the highwater mark of the Polytechniciens’ dominance. Analysts displaced the geometers. Among their successors was Pierre Laplace, for whom pictures were anathema and algebra was dogma.
And finally is the effect of very personal manners of thought. In 1925, when looking at the proof sheets for an article from a young Werner Heisenberg, something in particular got Dirac’s attention:
In the course of his calculations Heisenberg had noted that there were certain quantities for which A times B was not equal to B times A. Heisenberg was rather concerned by this peculiarity. Dirac seized on it as the key to the departure of quantum physics from the classical world. He believed that it was precisely in the modification of this mathematical feature that Heisenberg’s achievement lay. It may well be, as Darrigol, Mehra, and Rechenberg have argued, that the very idea of a multiplication that depends on order came from Dirac’s prior explorations in projective geometry.
Galison wants to understand:
the historical production of a kind of reason that comes to count as private
or to answer the question:
how was geometry infolded to become, for Dirac, quintessentially an interior form of reasoning?
He offers this description of Dirac’s later career and exclusively private use of projective geometry:
When Dirac moved to Cambridge to begin studying physics, he took with him this projective geometry and used it to think. But that thinking had now to be conducted only on the inside of a subject newly self-conscious of its separation from the scientific world. Dirac’s maturity was characterized again by flight, this time to Heisenberg’s algebra, an antivisual calculus that at once broke with the visual tradition in physics and with the legacy of an older school of visualizable, intuition-grounded descriptive geometry. With an austere algebra and Heisenberg’s quantum physics, Dirac stabilized his thought through instability: working through a now infolded projective geometry joined by carefully hidden passageways to the public sphere of symbols without pictures.
This is a very nice, novel perspective on the evolution of ideas that necessarily promotes the feeling that mathematics is deeply human and sometimes very personal.
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