As I write, there are upwards of 264 exabytes of digital storage, communication, and computation capacity available on the planet - in hard drives, servers, memory cards, flash drives, and other new media.68 That is a number with twenty zeroes and one that
exceeds the estimated number of grains of sand on the earth by more than 300 percent. This digital capacity exceeds our ability to store and circulate information using other media, like paper, by orders of magnitude. This is one reason why we are so often said to live in a digital age. A great deal of our world is housed in digital media.
These bytes and their constituent bits are used to encode, store, and manipulate all kinds of things from avatar gaming characters, photographs, and texts to scientific
68This number was put forward in 2011, and describes estimated digital storage capacity - including
hard drives, memory cards, DVDs and so on, up to 2007. Martin Hilbert, Priscila Lopez, “The World’s Technological Capacity to Store, Communicate, and Compute Information” inScience Vol. 332, No. 60 (April, 2011): 60 - 65, p. 60. This number was calculated using the modern standard in which each byte consists of eight bits.
models and massive databases. But on their own, bits are not meaningful. Bits don’t tell us anything or do any work for us unless we know how they are being used to encode photos, texts, models, or anything else. In order to make computers useful for a given task, communities of users, programmers, and software designers must first develop ways of reformulating that task and its constituent elements so that they can be stored, accessed, and controlled in and by the computer. We only live in a digital age insofar as people have devised ways of storing and accessing the things they care about using digital media. As Michael Mahoney put it, the history of computing is in part a history of how different communities have “put their portion of the world into the computer.”69 And there is never any single way that communities must digitize
their world. Instead, each community crafts or selects a particular set of encoding tools with which to translate pieces of their world into computational objects.
This chapter explores how one community put a small piece of the world of math- ematics “into the computer.” In particular, I investigate the development of an au- tomated theorem-proving program called the Logic Theory Machine that ran on the Johnniac mainframe at the RAND Corporation in Santa Monica, California in the mid-1950s. The Logic Theory Machine was designed to produce proofs of theorems taken from the pages of an early twentieth-century canonical text in elementary math- ematical logic - Principia Mathematica. In order to make this automation project possible, researchers at RAND transformed the elements of logic and the processes of theorem-proving into computational artifacts and operations.
I ask - what kind of work was involved in this transformation? What motivated this automation attempt? What kinds of obstacles and possibilities informed the process? How did these practitioners understand proof and think about mathematics in the
69Michael Mahoney, “What makes the history of software hard” in IEEE Annals of the History of Computing Vol. 30, No. 3 (2008): 8 - 18, p. 8.
context of computation? Ultimately, this chapter aims to recover and reconstruct new ways of representing the elements of logic that were devised in order to transport the world of Principia into the digital media of the Johnniac mainframe.
At bottom, this chapter is about the materials with which the work of mathematics is done. Traditionally, mathematics has been characterized as dealing with highly abstract and immaterial things. In fact, throughout history myriad material tools have been developed to make mathematics possible. From systems of written symbolic notation and diagramming to physical models and mechanical calculators, different technologies have equipped the heads and hands of mathematicians to formulate and explore their domain in different ways. Indeed, the world of Principia Mathematica
that was “put into the Johnniac” in the context of the Logic Theory Machine was not simply an abstract immaterial world of logic that lay in wait of representation. It was a world onpaper - a book world in which proofs were constructed on the page, typeset, bound, and circulated to communities of reading mathematicians.70
This chapter tracks a transformation from a human-oriented representational sys-
70In imagining the paper world ofPrincipiaI have in mind the rich and exciting scholarship that ex-
plores media, print, communication, and publication in history of science. I am particularly indebted to conversations with Alex Csiszar from whom I have learned a great deal about cultures of print, publication and the classification of mathematical knowledge at the turn of the twentieth century. See Adrian Johns,The Nature of the Book: Print and Knowledge in the Making (University of Chicago Press: 2000); Jim Secord,Victorian Sensation: The Extraordinary Publication, Reception, and Secret Authorship of Vestiges of the Natural History of Creation (University of Chicago Press: 2003); Lisa Gitelman,Scripts, Groves, and Writing Machines: Representing Technology in the Edison Era (Stan- ford University Press, 2000). Some scholars are interesting in looking at digital storage and recording mechanisms as episodes in history of writing - taking seriously the metaphor that information is “written” to discs and drives. I follow them in exploring digital storage as an episode in mathematical inscription and representation. In this regard, I follow media theorist Matthew Kirschenbaum,Mech- anisms: New Media and the Forensic Imagination (The MIT Press, 2012) - he treats digital storage as a form of literal writing. More I am eager to explore new forms of paper and pencil writing and diagramming that were developed in the process of designing and implementing computer programs. I was excited to learn of similar work from like-minded historians of computing Mark Preistly, David Norfre, and Gerard Alberts in “When Technology Became Language: The Origins of the Linguistic Conception of Computer Programming, 1950 - 1960” inTechnology and Culture,Vol. 55, No. 1 (Jan- uary 2014): 40 - 75, who explore different types of writing and representation in early programming practice. I explore some of these ideas in Dick, “Machines Who Write,” inIEEE Annals of the History of Computing, Vol. 35, No. 2 (April-June 2013): 85 - 87.
tem to a machine-oriented one, keeping an eye on where and how materiality matters for questions of mathematical agency and knowledge-production. In the context of
Principia Mathematica, logical propositions were inscribed on the page using a symbol system intended explicitly to capitalize on the human powers of vision and pattern recognition. I begin the chapter with an exploration of this paper world. The rest of the chapter tracks how new ways of representing logical propositions were developed - ones that accommodated a computer rather than a human practitioner.
In the context of the Logic Theory Machine, logical propositions were transported into the magnetic drum and core magnetic storage systems of the Johnniac mainframe in the form of what were called “Linked List Information Structures.” These were an early example of what we now call “data structures” - one of the tools computing practitioners have devised to “put their portion of the world into the computer.”71 They
are ways of organizing information in computer memory, of encoding things as digital things, of assigning meaning to underlying bits, of making digital media do work for us. Linked lists were crafted to overcome the significant memory management issues that plagued early computing machines whose storage capacity was very limited. Linked lists were one new form of materiality and representation designed to make logical propositions into digital things.
Anthropologist Jack Goody asked - “What’s in a list?”72 He proposed that the
evolution of human thinking was tied to the development of literary technologies like tables, charts, and lists. He wanted to know what ways of thinking, what forms of
71Where now “data” occupies a central position in our computational vocabulary, the word “infor-
mation” used to be the catch-all for what was inside computers and what they were representing in the world. The transformation from the language of “information” to the language of “data” hasn’t been thoroughly historicized. Relevant works include Thomas Haigh “Inventing Information Systems: The Systems Men and the Computer, 1950 - 1968” inThe Business History Review, Vol. 75, No. 1 (Spring 2001), pp. 15 - 61; Daniel Rosenberg, “Data before the Fact,” in“Raw Data” Is an Oxymoron, ed. Lisa Gitelman (Cambridge, MA: The MIT Press, 2013): 15- 39.
72Jack Goody, The Domestication of the Savage Mind (Cambridge, UK: Cambridge University
social organization, and what relationships to temporality and history were made and made possible with the development of written lists. I ask - “What’s in a linked list?” Where did that structure come from and what were its consequences for the material history of mathematics? What and how do linked lists represent? To whom or to what?
Linked lists are a window into the kind of work that was involved in putting mathe- matics into the computer. Part of this story is about the displacement of paper and of the human hand - a displacement ofPrincipia and the moment in the material history of mathematical practice that it embodies. However, as we will see in the concluding section of this chapter - the computer seldom replaces paper and it never replaces the human hand. Rather, computingdisplaces paper,displaces human labor, directs them towards different goals and problems.