By Arnold Serapilio
As a young man growing up in Cleveland, Ohio, Scott spent much of his time learning about computer programming and discovered it is concentric with mathematics. The designs of both studies are ordered by hierarchy, i.e., the approach to and execution of programming a computer and solving a math problem are essentially the same, so an understanding of mathematics is helpful when programming computers.
SG: A very simple sequence of numbers called the Farey sequence.
[The Farey sequence is the arrangement of irreducible fractions (in order of increasing size) between 0 and 1 with denominators less than or equal to given value n. E.g.: when n= 5, the Farey sequence is 0, ⅕, ¼, ⅓, ⅖, ½, ⅗, ⅔, ¾, ⅘, 1.]
John Farey was a British geologist who published a short letter in the May 1816 issue of Tilloch’s Philosophical Magazine and Journal inquiring about a mathematical property of the sequence. He has ever since been taken to task in the mathematics literature for stealing the insight from "somebody named Haros" so I decided to write a a book about Haros. As it turns out, it was a lion of French mathematics and a colleague of Charles Haros, Augustin-Louis Cauchy, who misnamed the sequence after Farey. See Stigler's Law of Eponymy: No scientific discovery is named after its original discoverer. By the way, Stigler's law was first recognized by Robert Merton so it is an example of itself.
[Scott's not just being funny. Though it's unclear to me whether Stigler's Law is truly, serendipitously an example of itself or whether Stigler exercised a little dramatic license—after all, who says math types can't be creative? More on that later.]
One of my favorite stories coming out of my work with the Athenæum’s Books Borrowed registers is about the Josiah Parsons Cookes, Sr. and Jr. Josiah Parsons Cooke, Sr., was an Athenæum proprietor and a widely-respected Boston lawyer. Josiah Parsons Cooke, Jr. was "the first university chemist to do truly distinguished work in the field of chemistry" in the United States. An apocryphal story about Jr. is that his interest in chemistry was sparked by reading Edward Turner's Elements of Chemistry. Sure enough, consulting the books borrowed register we find that on June 17, 1842, Sr. borrowed "Terners [sic] Chemistry." Jr. was 15 years old at the time. Not only is Turner's book still on the shelf at the Athenæum but I like to imagine that it is the very copy that was borrowed by Sr. and started the illustrious career of Jr."
SG: The absolute end of mathematics is when the percentage of mathematics in the future is zero everywhere in our concentric circle landscape. The effective end of mathematics is when the rate of change of this value is zero everywhere in the landscape. In either of these senses, how close to the end are we today? Mathematics researchers are the people moving mathematics from the future to the past; i.e., decreasing the percentage of mathematics in the future. I think of those working in the inner circles as pushing the percentage of mathematics in the past upward and I think of those working in the outer circles as pushing the percentage of mathematics in the past outward. I also think that they are both having a harder time. That is, the rate of change both upward and outward is tending to zero. [I]t’s harder to know more about what you already know a lot about and easier to know more about what you don’t know much about.
[Scott stresses more than once (primarily due to my obtuseness) the idea that today's findings encapsulate everything previously discovered. One finds the sum total of information on a topic to be the most generalized and abstract, which ultimately may prove unhelpful to somebody looking for solutions to a real-world problem. "Mathematics is not a discipline that appreciates its own history," Scott notes. To a mathematician, today's findings are the most inclusive, so why bother tracing the history? That Scott rejects this way of thinking is refreshing and heartening. He mentions how great the Athenæum's de-acquisition policy is: that there is no de-acquisition. Once we've obtained a book we hold onto it, ensuring an expansive, alive collection.]
[There is also the notion of creativity in mathematics. Scott asserts there are many very bright people applying mathematics in creative ways in order to solve practical problems; the example he gives is of Oliver Evans, the inventor of the automated grist mill, trying to determine the minimum amount of resources and correct conditions needed to yield the greatest output. Mathematicians who discount yesterday's techniques (which are simple in presentation and more readily understood) applied today's findings (abstract, more abstruse) and arrived at an answer mill builders intuited was insufficient. The mill builders themselves applied some more fundamental ideas and yielded more profitable results. If the future of mathematics is uncertain, it's because we are getting mired in the abstract rather than re-appropriating what we already know in new and innovative ways.]
Suppose then that the rate of change in all directions goes to zero; that despite their best and most strenuous efforts, research mathematicians cannot budge the horizon between past and future; that we know all the mathematics we can know and will ever know. There are certainly mathematical lines of investigation that have reached this impasse. Why not the entire field?
[Our conversation ends because Scott has to run upstairs to the Athenæum Encyclopӕdists meeting that begins in minutes. Before we part ways, though, he takes me down to Lower Pilgrim to show off the math section. Cutter H. I watch him pulling the books down from the shelves and admiring them. There is love and attention and reverence in his fingers and in his voice. Each book is tactile history, weighty but within reach.]
Jensen, William B. "Physical Chemistry before Ostwald: The Textbooks of Josiah Parsons Cooke," Bull. Hist. Chem, v. 36, n. 1(2011), pp. 10--21.