Musing about the Oxford maths kit yesterday brings to mind an age-old question: Are those mathematical instruments a product of philosophy or empirical science?
That is to say: Do we invent mathematics, or do we discover it in nature?
The question came up in a book I've been reading, Rachel Hewitt's Map of a Nation, a history of the early days of the Ordnance Survey, the British mapping agency.
In the 1820s, the Survey came to Ireland, then a British colony, to map the country on the unprecedented scale of six-inches to the mile. The first step was triangulation -- fixing the exact position of dozens of "trig points" by throwing a web of sight-line triangles across the land, beginning with a carefully measured base line. Then the so-called "interior surveyors" would fill in the details.
The surveyors were generally imbued with the conviction of many Enlightenment philosophes that the empirical study of nature was the proper source of all reliable knowledge. Lines, circles, triangles were nature's language, to be discovered by interrogating the external world.
The great Anglo-Irish mathematician William Rowan Hamilton became much interested in the survey. He believed, however, that geometry was a branch of the philosophy of mind, "a product of the extraordinary powers of independent creation that were possessed by the intellect and which operated quite separately from the world of materiality and causation" (I quote Hewitt). He imagined when the first triangle was invented, like a light flashing on in the mind of an early philosopher. In this he followed Kant's suggestion that geometry was an image of the internal, not the external landscape.
One can imagine stimulating discussions between the mathematician Hamilton and the practical-minded leaders of the survey, Thomas Colby and Thomas Larcom.
Of course, the successful completion of the survey did not depend upon resolving the philosophical question, which continues to be debated today. For myself, it seems rather a chicken and egg sort of question, given that the human mind is itself a product of nature. What is truly interesting, and still deeply mysterious, is the power of the mind to abstract the general from particulars.
Those instruments in the student maths kit represent abstractions, only approximately discerned in nature -- as the heroic labors of the Ordnance surveyors to achieve perfect accuracy made abundantly clear.