In the section “Proving Theorems in the Twentieth Century” in the Introduction to this dissertation, I introduced Whitehead and Russell’s canonical textPrincipia Math- ematica. It was part of a particular mathematical tradition that sought to reduce mathematics to logic. Whitehead and Russell aimed to craft formal systems consisting of sets of axioms and rules of inference for deriving consequences from them.73 Their
project reflected a particular philosophy of mathematical knowledge - that it could be fully formalized, axiomatized, and standardized within mathematical logic. For them, a proof consisted of a chain of deductive steps that began with the axioms of their logical system and concluded with a true and interesting logical statement, and they believed the whole of mathematical knowledge could be formulated in this way.
However, the project also demonstrated a commitment to a particular medium as the site where mathematical knowledge should be produced and circulated - namely, paper and the book. Proofs were not just sequences of deductive steps in the abstract, but literally written sequential lists of logical propositions, each step marked by the principle of inference that permitted one to follow from the next. Mathematical knowl- edge was not hovering in the platonic ether, it was collected, catalogued, and circulated as lists of theorems, lists of inferences, and lists of proofs on the page and collected in the three volumes of Principia were first published by Cambridge University Press in 1910, 1912, and 1913.
In order to equip the heads and hands of mathematicians to do the work of proof, on paper, within their logical system, Whitehead and Russell also crafted a particular
notational system - a way of writing, representing, and exploring logic on the page. Proof would be standardized not just formally but symbolically as well.74 In their
account, the notation was not merely an incidental or arbitrary tool used to access the truths of logic, but a necessary condition for the exploration of logic in the first place:
The symbolic form of the work has been forced upon us by necessity: with- out its help we should have been unable to perform the requisite reasoning. It has been developed as the result of actual practice, and is not an ex- crescence introduced for the mere purpose of exposition. [...] No symbol has been introduced except on the ground of its practical utility for the immediate purposes of our reasoning.75
They started with the notational system developed by Peano - an Italian mathemati- cian who worked on the project of providing an axiom system for the natural numbers76
74A helpful introduction to the notational system in Principia, including a compari-
son with more contemporary logical notation is available in Bernard Linksy, “The No- tation in Principia Mathematica” in Stanford Encyclopedia of Philosophy (Fall 2011): http://plato.stanford.edu/archives/fall2011/entries/pm-notation/.
75Whitehead, Russell, Principia, viii.
76See, for example, Peano, [1889] "The principles of arithmetic, presented by a new method" in
Jean van Heijenoort,A Source Book in Mathematical Logic, 1879–1931. (Cambridge, MA: Harvard University Press, 1967): 83–97.
- and praised him for showing “how symbolic logic was to be freed from its undue ob- session with the forms of ordinary algebra, and thereby made it a suitable instrument for research.”77 Algebra was typically interpreted such that the symbols and variables
stand in for unknown numerical values or quantities, whereas the symbols of logic in Peano’s rendering and Russell and Whitehead’s work are taken to stand in for proposi- tions or entities of any kind, making it a more powerful language. Part of the project of thePrincipia was to establish this more powerful potential of symbolic formalism, and to enlarge the scope of algebra-like symbol systems beyond the numerical domain.78
In the Introduction to Principia, they go to even greater lengths, emphasizing the importance of their notational system for the work of mathematical logic by offering five justifications of their symbol system in relation to the limitations of both natural language and numerically-bounded symbol systems. Of particular interest here are the third and fourth reasons: “The adaptation of the rules of the symbolism to the processes of deduction aids the intuition in regions too abstract for the imagination readily to present to the mind the true relation between the ideas employed.”79 That
is to say, the symbol system makes possible more abstract cognition and imagination than natural languages while still capitalizing on the sequential, left to right form of
77Whitehead, Russell, viii.
78There is an interesting parallel for this shift in history of mathematics from numerical to non-
numerical symbol systems in history of computing. As discussed in “The Possibilities of Computing” in the Introduction, automated theorem-proving practitioners were among those who wanted to use computers to perform nonnumeric tasks, where they were first used primarily for calculation. Both historical moments call for a new way of mobilizing a set of tools - written symbolism in the former and computing machinery in the latter - for new domains. Both begin with a set of tools crafted for numerical work, and fashioned new uses and new interpretations of them that enabled additional domains of application. Both transformations also have ontological stakes concerning what a formal systemis - be it deductive or algorithmic. Both answer that formal systems are not limited to numer- ical domains. Natural language, cognition, medial diagnosis, face recognition - all manner of domains - are then opened up to representation and exploration with formal tools like that of algebra or cal- culation, de-numericized as logic and computation. However, bringing formal tools to bear on these many domains transforms them - they are understood and practiced differently as they are fashioned in terms of deduction or algorithm.
natural language for readability.
They go on to attribute this potential precisely to the visual properties of the symbol system - which renders abstract ideas and relations concretely and succinctly to the human eye: “The terseness of the symbolism enables a whole proposition to be
represented to the eyesight as one whole, or at most in two or three parts... This is a humble property, but is in fact very important in connection with the advantages” of the symbol system for the intuition of very abstract objects.80 The centrality of
symbolism in mathematical logic was also emphasized by Peano and Frege before them and David Hilbert and Wilhelm Ackermann after them. Hilbert and Ackermann, for example, wrote that their logic employs “a symbolic language like that which has long been in use to express mathematical relations. [...] The great advances in mathematics since antiquity, for instance, in algebra, have been dependent to a large extent upon success in finding a usable and efficient symbolism.”81
Principia thus combines a formal system (propositional logic), a medium (paper and the book), and a way of writing mathematics (their notational system) to embody the turn of the twentieth century logicist vision of mathematics. As seen in Figure 1.1 and Figure 1.2, the notational system was both a handwritten part of the production of Principia and was reproduced in its typesetting for publication.
On those pages, each line is created by the application of an accepted rule of inference to the proposition on the line before. Each line contains a single proposition with implication, written as , as the main logical operator. The left hand implies the right hand. The topmost statement is the theorem to be proved. The proof follows,
80Whitehead, Russell, Principia, 3, my emphasis. The other justifications can be paraphrased as
follows: 1) Natural language is insufficiently abstract and precise for the work of mathematical logic; 2) The grammar of natural language is too flexible for mathematical logic; 5) In order to attain the “complete enumeration of all the ideas and steps in reasoning employed in mathematics,” a formal, rigorous symbol system is required. The justifications are enumerated on pp. 2 - 3.
Figure 1.1: Manuscript leaf from Principia Mathematicia. MU-BR, c.a. 1907.
and is meant to be read from top to bottom. This kind of symbolic system is taken largely for granted today, but it is worth noting that it has a particularspatiality and
materiality and that these have built in assumptions about mathematical agents and mathematical practice.
Principia represents a particular moment in the history of mathematical writing, mathematical materiality, and mathematical practice.82 Here, the agent of proof is
assumed to be a reasoning, seeing, writing, reading person and the notation system is developed accordingly. Material representational systems are tools for thinking, for making, and for communicating mathematical knowledge. The notational system with which Russell and Whitehead make the propositional logic manifest in their work is neither secondary to the project, nor arbitrarily construed. Rather they understood the notation as the enabling condition for constructing and cognizing the deductive system upon which they wanted to ground all of mathematics. It was not possible to transport this written system, unaltered, into the context of computation. A different
82The relationship between notational systems and cognition has recently become a subject of
interest for science studies scholars. For example, Ursula Klein has explored the role of “paper tools” in the knowledge-practices of nineteenth century organic chemistry in Experiments, Models, Paper Tools: Cultures of Organic Chemistry in the Nineteenth Century. (Stanford, CA: Stanford University Press, 2003). Hans Jörg Rheinberger makes a compelling case that way in which scientists take notes during their experimental work shapes the conclusions they will later make concerning what that work was about and what it revealed. See in particular, “Scrips and Scribbles” inMLN, Vol. 118, No. 3 (April, 2003): 622 - 636. Bruno Latour suggests that a study of the making, circulation, and reproduction of “inscriptions” would go a long way to understanding knowledge-production in for example, “Visualization and Cognition” inKnowledge and Society: Studies in the Sociology of Culture Past and Present, eds. Henrika Kuklick and Elizabeth Long (Jai Press Inc., 1986): 1 - 40. An interest in notational systems has also become an interest for certain historians of mathematics. Notably, the work of Reviel Netz on origin of deductive reasoning practices with the lettered diagram and certain linguistic formulations in ancient Greek mathematics. See, for example, Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Cambridge, UK: Cambridge University Press, 1999); Netz, “Linguistic Formulae as Cognitive Tools” inPragmatics and Cognition, Vol. 7, No. 1 (1999): 147 - 176. Rotman, as discussed earlier in Part One, also has a pointed interest in mathematical notation systems in so far as he wants to reduce mathematics to writing
and associated semiotic practices. See especially Rotman,Mathematics as Sign: Writing, Imagining, Counting (Stanford, CA: Stanford University Press, 1993); and Rotman, “Thinking Dia-Grams : Mathematics and Writing” in ed. Mario Biagioli, Science Studies Reader (New York: Routledge, 1999): 430 - 441.
notational system and representational structure would be needed ifPrincipia was to be automated. It would have to accommodate quite a different mathematical agent, abled and limited in significantly different ways than the imagined readers of Principia. I now turn to the development of Herbert Simon, Allen Newell, and John Clifford Shaw’s Logic Theory Machine to explore how that was done.