Think Piece

What’s in a Name? Anonymity in the Authorship of Babylonian Scientific Texts

By E.L. Meszaros

What does ancient scientific authorship have to teach about the methods and practices of science? Ancient cultures in what is now Iraq and Iran carried out substantial scientific investigations into areas of math, astronomy, medicine, and countless other modern disciplines. We have written works — tablets impressed with the wedges of cuneiform writing — recording everything from astronomical observations to legal agreements. Contrary to modern expectations, however, these texts are often not associated with an author but instead circulate(d) anonymously. 

Even when mentioned, Babylonian authors occupy a marginal space in the arrangement of text, identified in colophons rather than on title pages or in neat bylines. These small notes tacked on to the end of texts paint a complex idea of the relationship between authors and texts, often identifying both a scribe and an owner of the tablet. The culture of scribal copying and the different ways that individuals could contribute to a text have all supported ideas of collaborative authorship and allow us to think of a text as an “ongoing, contributive enterprise” (Foster, “On Authorship in Akkadian Literature”). Unlike contemporary research groups, this collective work would be continued, copied, and modified for generations, with each scribal hand a work passed through potentially contributing content, style, and even critique. This is especially evident in the observational program of the Astronomical Diaries, which went on for hundreds of years and must necessarily have been the collaborative work of many different scientists.

Image of the reverse of Tablet K 75 from the Cuneiform Commentaries Project. The Colophon is the last four lines, all beneath the horizontal ruling and visually removed from the spread-out text above.

We can identify authors of these tablets in other ways as well. Notably, authors sometimes inserted their own names as acrostics, using the first syllabic sign of each line to spell out their name. Assyriologists like to talk about the attributed “divine authorship” of texts as well, an idea which certainly complicates notions of who an author is. Regardless of how you view the idea of divine authorship, the notions of scribe, owner, collective observational programs, and acrostics all point to an idea of authorship that differs from our modern conception.

But the larger problem isn’t just that Babylonian and modern authorship differ. Rather, we must contend with the fact that in most cases we have no idea who the authors of any given Babylonian scientific text actually are. While Assyriologist W. G. Lambert writes that the “overwhelming majority of Babylonian texts circulated anonymously,” Babylonian scientific texts are perhaps particularly characterized, according to Markham Geller, by their anonymity. But the term “anonymous” carries a lot of weight in the modern context, when it is a specific choice or statement meant to indicate something about the content of a piece.  When we talk about anonymous Babylonian scientific texts, however, “anonymous” could be viewed as the default form of authorship.  Relying on modern conceptions of anonymity might muddy the truth of what’s actually happening with authorship of Babylonian scientific texts and how we should think about attributing knowledge production in the ancient Near East.

Anonymity of Babylonian Scientific Texts

Assumptions around anonymity are entangled with research methods and play a role in assigning value to documents that attest to Babylonian scientific culture. The modern value of authorial attribution appears in questions of priority, asking who discovered or invented an idea first. Priority questions have plagued the history of science, and they make sense in a culture where the identification of authorship and association with new developments is privileged. We also see modern standards of authorship at play in narratives of genius, where large-scale steps or important contributions are attributed to one person. Priority and genius narratives are problematic for a modern history of science, but become even more of an issue when brought back to the study of Babylonian scientific texts, where cultures of anonymity and authorship differ substantially from modern ideas.

We can see ideas of modern scientific authorship at odds with Babylonian anonymity in how certain ancient texts and methods have been treated by modern scholars. Take, for example, Tablet YBC 7829. This mathematical text, from the Yale Babylonian Collection, shows a square crossed by two diagonal lines along with some cuneiform writing. Outside the square are 3 v-shaped wedges indicating that the sides of the square measure 30. The center of the square contains two lines of numbers, the first providing a sexagesimal value for the square root of 2, and the second 30 times that value.

Image to YBC 7829 Tablet from the Cuneiform Digital Library Initiative

Those familiar with geometry may recognize this as an example of the Pythagorean theorem at work. Perhaps unsurprisingly attributed to the Greek philosopher and mathematician Pythagoras, this theorem defines the square of the length of the long side of a triangle as the sum of the square of its sides. It may be more familiar as:

a² + b² = c²

Or, in the case of this tablet, 30² + 30² = x², which we can solve as 1800 = x², giving us 30√2 = x. (Although, of course, reducing YBC 7829 to such a formula is disingenuous.)

The problem of anonymity becomes evident with this application: can we ask if people, centuries before Pythagoras, were using “Pythagoras’ theorem?” In the absence of an identified author on this tablet, however, what is the best way of contesting this strange priority-assertion? The anonymous nature of YBC 7829 contributes to the treatment of this tablet in Greek terms. Similarly, since the irrational square root of 2 proved so important in the history of Greek mathematics, many scholars placed undue importance on its representation on Babylonian tablets. YBC 7829 is held as a special example of unique knowledge, when in reality the tablet is likely nothing more than an exercise done by a trainee scribe. The weight and importance of representing √2 doesn’t come from Babylonian mathematical culture, but rather from ours.

Image of Plimpton 322 Tablet from the Cuneiform Digital Library Initiative

We have further evidence that Babylonian mathematicians understood this relationship between the lengths of the sides of a triangle. Plimpton 322 shows a table of numerical cuneiform data that fit the Pythagorean Theorem. Often called “Pythagorean Triples,”  these numbers are integer solutions for a, b, and c in the formula a² + b²= c². It is clear from these examples that Babylonian scientists — mathematicians, if you feel the need to disambiguate — understood the relationship between the lengths of a triangle’s sides.

The question of priority plagues the analysis of these tablets, however, especially in the eyes of the modern media who cannot seem to stop asking which culture had these ideas first.  Their approach frames an understanding of Babylonian math and science with Greek culture, forcing us to examine Babylonian contributions through a Greek lens. But this view is artificial and necessarily a-historic, and is aided by an inability to push back with a Babylonian alternative to Pythagoras due to the anonymity of these ideas. These tablets suggest that a different approach to knowledge is possible, one that engages with scientific discourse without recourse to self-identification. As such, these contributions are  better understood when removed from the framework of “Pythagoras,” “trigonometry,” and priority in general. 

Priority inserts itself into the study of Babylonian science in other ways. Systems A and B are two different methods for calculating when a phenomenon like the first visibility of a planet will occur and where it will be located in the sky when it does. System A, usually called a “step function,” adds constant values to a time and location to find the next time and the next location. Most planets have a few constant values, depending on what part of the sky they are in. System B, the “zig-zag function,” adds a constantly changing value to previous time and location. 

The complexity of these Systems suggested to some scholars that they were the creation not of the collaborative efforts of many astronomers over a period of time, but rather the inventions of a genius. For some decades, scholars sought to identify these “geniuses” and attribute authors to both of these systems, landing on Nabû-rēmanni for system A, and Kidinnu for system B. Each of these names  is supported by an appearance in a tablet’s colophon (Neugebauer, “The Alleged Babylonian Discovery of the Precession of the Equinoxes”), though, importantly, there is very little evidence for either name. 

The most interesting aspect of these authors is how they are identified. Rather than the expected scribe or tablet owner, colophons have identified texts as “computed table of” Kidinnu and Nabû-rēmanni (Ossendrijver, Babylonian Mathematical Astronomy). This led to the theory that Kidinnu and Nabû-rēmanni  are instead the contributors of the theory used to create the data contained on the tablet — the “geniuses” responsible for Systems A and B. This focus is misplaced, however, in that it asks us to accept not only poorly attested identities of these “geniuses” but also to look for attested creators within a culture that does not seem to have privileged such information. Questions of priority and genius falter against the anonymous contributions of the ancient Near East — not only are they hard to answer but they might be the wrong questions to ask.

Disentangling Ideas of Authorship

The search for priority in the Babylonian scientific context is often a-historical. Worse, by seeking out individuals to credit with entire theories, our modern biases often influence who we look for in the historical record, often seeking out upper class, male contributors. Pushing back against the genius, individual author and looking instead for collaborative work taking place over time may create more space for women in the authorship of Babylonian science. We know, for instance, that women could be astronomers and scholars, knowledgeable in cuneiform writing, even though most of the recorded Babylonian “authors” were male. Relying only on these noted “scribes” and “owners” erases the role that women may have played in observational programs and the long collaborative and contributive enterprise of Babylonian authorship.

The search for “authorship” of Babylonian science texts is flavored by modern conceptions of how authorship should work. Naive ideas that priority matters, that work should be associated with individuals rather than communities, and that anonymity is rare and, thus, inherently meaningful, all distract from understanding authorship as Babylonian scientists would have. We should acknowledge the complexity of the modern author while working to disentangle modern meanings from ancient ideas of authorship. We can then complicate the narrative of scientific authorship in both directions, enabling historians and historians of science to reshape how we think and talk about authors, but also encouraging a less-broad view of modern discussions of scientific authorship. Just as we confront changing ideas of what “science” means, we must also contend with shifting ideas of what it means to author scientific knowledge.

Scientific Anonymity in the Modern Age

The institutions of (modern) science privilege originality and require named contributions. Anonymity, when it’s used, is a conscious choice. Mary Terrall, in “The Uses of Anonymity in the Age of Reason,” describes how the anonymity of the Paris Academy of Sciences prize allowed the Marquise de Châtelet to submit her work without declaring her gender, a boon in times when women were not often allowed in scientific circles. However, when she did not win she gave up her anonymity to use her station and powerful connections to get her work published. 

This shifting use of anonymity suggests some key features of modern anonymous authorship in scientific texts — it was almost always temporary and intended to eventually be breached. Terrall explains that almost without exception “anonymous authors of texts on scientific subjects either unmasked themselves or were unmasked by others within a few years, if not months, of publication.” In this sense, we can talk about modern anonymity in scientific authorship as an unusual and temporary state worthy of note.

Anonymous science becomes even rarer in the present, with professional advancement in labs and academia dependent on evidence of publication. Modern authorship in science becomes a power statement, with the order of authors as they are listed in an article or book frequently indicative of power dynamics within partnerships. The first author position is usually granted the most acclaim, but the last author is usually the person with the most power — the project or lab leader, for example. Such power dynamics are unlikely components of Babylonian scientific texts, but are important components for understanding modern scientific authorship.

We also see anonymity in the authorship of reviews and critique. The anonymous peer review process has become a staple for the publication of research, an apparatus not limited to scientific discourse but certainly foundational to the modern practice of science. Concealing the identities of both a paper’s author and its reviewer is designed for protection and to increase honesty and objectivity, yet often has the effect of reducing credibility and accountability. In fact, while anonymity in the context of peer-review is industry standard, its value and effectiveness have been questioned and found wanting.

Modern scientific authorship is complex — it must necessarily deal with ideas of precarity and the power dynamics at play in how authorship is recorded. But more than this, it has become so ingrained within the institution of science that concepts like the“number of first-authored papers” become important. Coupled with modern conceptions of anonymity in scientific texts, we have a view of how scientific authorship works that is multifaceted and also not always consciously enacted. This is the baggage that we carry with us when we try to examine authorship and knowledge in ancient science.

Associating written works with a specific author (or authors) is second nature in the modern era. It’s not only safe to assume that a piece of recorded knowledge can be attributed to a creator of that knowledge, but practices of citation and promotion require us to acknowledge these creators. Babylonian approaches to scientific authorship prompt us to question our modern ideas of what an author is and why attributive authorship is necessary. Rethinking scientific authorship to incorporate long-term, collaborative enterprise and unremarkable anonymity allows us to better understand ancient scientific texts. It also forces us to confront uncomfortable questions about modern scientific texts, like who gets to claim authorship and who can or must make use of anonymity. These structures are broader and deeper than our modern conceptions, stretching back thousands of years and encompassing necessity, precarity, and manifestations of power in the production of scientific knowledge. 

E.L. Meszaros is a PhD student in the History of the Exact Sciences in Antiquity at Brown University. Her research focuses on the language used to talk about science, particularly as this language is transmitted between cultures and across time.

Featured Image: Stone panel from the South-West Palace of Sennacherib, The British Museum.

Intellectual history

Genres of Math: Arithmetic, Algebra, and Algorithms in Ancient Egyptian Mathematics

By contributing author E.L. Meszaros

As non-native readers of Egyptian hieratic and hieroglyphics, our understanding of the mathematics recorded in these languages must necessarily go through a process of translation. Such translation is both necessary to allow us to study these problems, but also precarious. If done improperly, it can prevent us from true understanding. One way that we approach translating Egyptian math problems is by grouping them into genres, using categorization to aid in our translation by thinking about problems as algebraic or geometric equations, crafting them into algorithms, or piecing together word problems from their prose. If the process of understanding Egyptian math problems relies so heavily on translation, and translation in turn is influenced by categorization, then we must consider how our processes of categorization impact our understanding of ancient Egyptian math. 

The necessity of translation for the modern study of ancient mathematics has been the source of a great schism within the community. In an infamous 1975 paper, Unguru argued that one of the unintentional consequences of translation was the attribution of algebraic thinking to these ancient cultures. Mathematicians and historians tend to translate the word problems of ancient Iraq or Egypt into the abstracted symbolic statements we are familiar with today. This has helped us to better understand ancient mathematical ideas, but has also done a disservice to the math itself. The process of abstraction manipulated the geometry or arithmetic of ancient math into algebra, a way of examining mathematical problems that Unguru argued these ancient cultures never used (78).

Image of a fragment of the Moscow Papyrus showing problem 14 on how to calculate the volume of a frustum. The top portion shows the original hieratic, which has been translated below into Egyptian hieroglyphics.

However, others have pushed back against Unguru. Van der Waerden suggests that Unguru has misunderstood “algebra” by attributing such importance to the symbolic representation of data. Rather, van der Waerden emphasizes the convenience of symbols as a way of interpreting, analyzing, and comparing data, rather than the structural language of understanding data (205). Freudenthal similarly takes umbrage with Unguru’s understanding of what algebra is. “Symbols,” he writes, “…are not the objects of mathematics…but rather they are part of the language by which mathematical objects are represented” (192).

We can compare the strict translation of an Egyptian word problem to its algebraic translation by looking at problem 14 of the Moscow Papyrus.

Prose English translation:
Method of calculating a / ̄\.
If you are told a / ̄\ of 6 as height, of 4 as lower side, and of 2 as upper side.
You shall square these 4. 16 shall result.
You shall double 4. 8 shall result.
You shall square these 2. 4 shall result.
You shall add the 16 and the 8 and the 4. 28 shall result. 
You shall calculate  ̅3 of 6. 2 shall result.
You shall calculate 28 times 2. 56 shall result.
Look, belonging to it is 56. What has been found by you is correct. (Translation by Imhausen 33)

Algebraic Translation:
V = 6 (22 + (2*4) + 42)/3

Abstracted Algebraic Translation:
V = h (a2 + ab + b2)/3
h (height) = 6
a (base a) = 2
b (base b) = 4
V = volume

The algebraic translations are at once easier to take in but also visibly shorter, clearly missing information that the prose translation contains.

As an alternative to these translation techniques, Imhausen proposes the use of algorithms. Imhausen suggests that we translate Egyptian mathematical problems into a “defined sequence of steps” that contain only one individual instruction (of the type “add,” “subtract,” etc.) (149). These algorithms can still represent math problems in multiple ways. A numerical algorithm preserves the individual values used within Egyptian problems, while a symbolic form abstracts the actual numbers into placeholders (152). 

Numeric Algorithmic Translation:

  1. 42 = 16
  2. 4 x 2 = 8
  3. 22 = 4
  4. 16 + 8 + 4 = 28
  5.  ̅3 x 6 = 2
  6. 2 x 28 = 56

Here the first three numerical values are the given bases and height from the problem. The unfamiliar ” ̅3″ is the standard way of writing a fraction of 3, namely 1/3, in ancient Egyptian math.

Symbolic Algorithmic Translation:

  1. D22
  2. D2 x D3
  3. D32
  4. (1) + (2) + (3)
  5.  ̅3 x D1
  6. (5) x (4)

Drawing out the scaffolding of the problem by defining such algorithms allows scholars to easily compare math problems. The abstraction into symbols, the removal of extraneous information, and the sequential rendering allow us to more easily notice variation or similarity between problems (“Algorithmic Structure” 153). Imhausen suggests that identifying the substructure encoded beneath the language of presentation allows us to compare individual math problems not only with each other, to generate groups of mechanisms for solving and systems of similar problems, but also to look cross-culturally. Breaking down problems from Mesopotamia, China, and India may reveal similarities in their underlying algorithmic structures (158). 

The generation of algorithmic sequences from Egyptian word-based math problems does not solve all of our translation problems, however. Any act of translation, no matter how close it remains to the original language, is a choice that necessitates forgoing certain options. It also allows for the insertion of biases on the part of the translator themselves—or rather, such insertion is unavoidable.

In the example from the Moscow Papyrus, for example, the initial given values of the frustum are not specifically identified. The images from the original problem are missing, as are the verbs for the mathematical operations. Imhausen herself points out that this algorithmic form reduces some interesting features. The verb “double” in the original problem, for example, is replaced with “x 2” in the algorithmic translation (75). Making these changes requires us to confront the choice between algorithmic structure and staying true to the source material. “Fixing” these differences allows us to more easily compare math problems, but also presumes that we know what was intended.

The translation of Egyptian math problems into schematic algorithmic sequences is, therefore, not without its own set of problems. While Imhausen claims that they avoid some of the pitfalls of translation into algebraic equations that have so divided the community (158), algorithm interpretations are still likely to present the material in a way that differs from how ancient mathematicians thought about their own material. However, when applied carefully, such mapping may provide valid interpretations of these texts and a focal point for comparison.

Thinking about the genre of translation, the use of algebraic or geometric or algorithmic tools to interpret ancient math, is important for a number of reasons. We have already seen that the choice of genre impacts ease of understanding. Modern scholars used to thinking about math problems in an algebraic format will, unsurprisingly, read algebraic translations more easily. But these choices also impact what aspects of the original we preserve — algebraic translations lose information about the order of operations and remove the language used to present the problem.

However, paying attention to generic classification can also prevent us from reading ancient math problems with the “Western” lens. While algebraic interpretations are an artifact of modern scholarship, they are also an artifact of European scholarship. Too often the idea of geometry is put forward as an entirely Greek invention, while algebra is thought of as belonging to Renaissance Europe. By privileging these ways of thinking about ancient math problems we may be inherently white-washing native Egyptian thinking. Prioritizing algebraic interpretations, even if they aid in understanding, work to translate Egyptian math into the more familiar “Western” vernacular. Instead, scholars should work with the unfamiliar and think about these math problems without filtering them through these modern concepts.

Regardless of who one sides with in the debate between algebra and arithmetic, prose and algorithm, we must be cognizant of the fact that categorizing ancient Egyptian math is a conscious choice that influences how these problems are understood. Much like the act of translation itself, categorization is a process that is inherently influenced by the biases—intentional or otherwise—of the scholar. There may be nothing wrong with thinking about Moscow 14 in terms of an algebraic equation as long as we understand that this is an act of translation from the original and, therefore, reflects a reduced understanding of the native problem itself and incorporates aspects of the translator’s biases.

Which is all to say: tread carefully, because even numbers are not immune to the bias of translation.

E.L. Meszaros is a PhD student in the History of the Exact Sciences in Antiquity at Brown University. Her research focuses on the language used to talk about science, particularly as this language is transmitted between cultures and across time.

Think Piece

The challenge of contingency and Leibniz’s cybernetic thinking

By guest contributor Audrey Borowski

Gottfried Wilhelm Leibniz, painted by Christoph Bernhard Francke

According to the philosopher of science Alexandre Koyré, the early modern period marked the passage ‘from the world of more-or-less to the universe of precision’. Not all thinkers greeted the mathematization of epistemology with the same enthusiasm: for the German philosopher Martin Heidegger, this marked a watershed moment when modern nihilism had taken root in the shape of the reduction of the world to calculation and recently culminated with the emergence of cybernetics. One of the main culprits of this trend was none other than the German mathematician and polymath Gottfried Leibniz (1646-1716), who in the late seventeenth century invented the calculus and envisaged a binary mathematical system. Crucially, Leibniz had concerned himself with the formalization and the mechanization of the thought process either through the invention of actual calculating machines or the introduction of a universal symbolic language – his so-called ‘Universal characteristic’– which relied purely on logical operations. Ideally, this would form the basis for a general science (mathesis universalis). According to this scheme, all disputes would be ended by the simple imperative ‘Gentlemen, let us calculate!’

A graphic representation of second-order cybernetics by Mark Côté

For having mechanized reasoning, cyberneticist Norbert Wiener touted Leibniz as a ‘patron saint for cybernetics’ (Wiener 1965, p. 12) in the ‘Introduction’ to his 1948 seminal work Cybernetics or Control and Communication in the Animal and the Machine. In it, he settled on the term ‘kybernetes’, the ‘steersman’ to describe a novel type of automatic and self-correcting reasoning which consisted in the deployment of mathematics, notably via a feedback mechanism, towards the domestication of contingency and unpredictability. Cybernetics does not ‘drive toward the ultimate truth or solution, but is geared toward narrowing the field of approximations for better technical results by minimizing on entropy––but never being able to produce a system that would be at an entropy of zero…. In all of this, [it] is dealing with data as part of its feedback mechanism for increasing the probability of a successful event in the future (or in avoiding unwanted events).’

Cybernetic applications are ubiquitous today from anti-aircraft systems to cryptography; an anti-aircraft system, for instance, receives input data on a moving target and delivers the navigation of bullet to the target as output after a computing process.  Cybernetics’ aim is first and foremost practical and its method probabilistic: through the constant refining of the precision of a prediction, it helps steer action through the selection between probabilities. Under those conditions, a constant process of becoming is subordinated to a weak form of determinism; real infinite complexity is deferred in favour of logical symbolism and ‘disorganization’, that ‘arch-enemy’ endemic to intense mutability as Nobert Wiener put it, gives way to ontological prediction.

In his works The Taming of Chance and The Emergence of Probability Ian Hacking traced the emergence of probabilistic thinking away from deterministic causation. In fact and against commonly-held positivist narratives of the triumph of objective rationality, historians of mathematics generally acknowledge that the seventeenth century witnessed the birth of both probability theory and modern probabilism perhaps most famously epitomized by Pascal’s Wager. With the emergence of contingency, the question of its conceptualization became all the more pressing.

Perhaps no thinker was more aware of this imperative than Leibniz. Leibniz is often portrayed as an arch-rationalist and yet he did not view pure deduction as sufficient for reasoning; the ‘statics’ inherent to his characteristic (Leibniz, 1677) were simply ill-suited to a constantly evolving practical reality. Finite calculation needed to be complemented by probabilistic reasoning (1975, p. 135) which would better embrace the infinite complexity and evolving nature of reality. Although the author of a conjectural history of the world, The Protogaea, Leibniz did not merely conjecture about the past, but also sought to come to grips with the future and the state of mutability of the world. To this end, he pioneered the collection of statistical data and probabilistic reasoning especially with regards to the advancement of the modern state or the public good (Taming of Chance, 18). Leibniz had pored over degrees of probability as early as his 1665 law degree essay De conditionibus and the ability to transmute uncertainty into (approximate) certainty in conditions of constant mutability remained a lifelong preoccupation. More specifically, he set out to meet the challenge of mutability with what appears as a cybernetic solution.

An example of Leibniz’s diagrammatic reasoning

In a series of lesser-known texts Leibniz explored the limits and potentially dangerous ramifications of finite cognition, and the necessity for flexible and recursive reasoning. In 1693 Leibniz penned The Horizon of the Human Doctrine, a thought experiment which he subtitled: ‘Meditation on the number of all possible truths and falsities, enunciable by humanity such as we know it to be; and on the number of feasible books. Wherein it is demonstrated that these numbers are finite, and that it is possible to write, and easy to conceive, a much greater number. To show the limits of the human spirit [l’esprit humain], and to know the extent to these limits’. Building on his enduring fascination with combinatorial logic that had begun as a teenager in 1666 with his De Arte Combinatoria and had culminated ten years later with his famous ‘Universal Characteristic’, he set out to ‘show the limits of the human spirit, and to know the extent to these limits’. Following in the footsteps of Clavius, Mersenne and Guldin, Leibniz reached the conclusion that, through the combination of all 23 letters of the alphabet, it would be possible to calculate the number of all possible truths. Considering their prodigious, albeit ultimately finite number, there would inevitably come a point in time when all possible variations would have been exhausted and the ‘horizon’ of human doctrine would be reached and when nothing could be said or written that had not been expressed before (nihil dici, quod non dictum sit prius) (p. 52). The exhaustion of all possibilities would give way to repetition.

In his two later treatments on the theme of apokatastasis, or ‘universal restitution’, Leibniz took this reasoning one step further by exploring the possible ramifications of the limits of human utterability for reality.  In them, he extended the rule of correspondence between possible words to actual historical events. For instance, since ‘facts supply the matter for discourse’ (p. 57), it would seem, by virtue of this logic, that events themselves must eventually exhaust all possible combinations. Accordingly, all possible public, as well as individual histories, would be exhausted in a number of years, inevitably incurring a recurrence of events, whereby the exact same circumstances would repeat themselves, returning ‘such as it was before.’ (p.65):

‘[S]uppose that one day nothing is said that had not already been said before; then there must also be a time when the same events reoccur and when nothing happens which did not happen before, since events provide the matter for words.’

In a passage he later decided to omit, Leibniz even muses about his own return, writing once again the same letters to the same friends.

Now from this it follows: if the human race endured long enough in its current state, there would be a time when the same life of certain individuals would return in detail through the very same circumstances. I myself, for example, would be living in a city called Hanover situated on the river Leine, occupied with the history of Brunswick, and writing letters to the same friends with the same meaning. [Fi 64]

Leibniz contemplated the doctrine of Eternal Return, but it was incompatible with his metaphysical understanding of the world. Ultimately, he reasserted the primacy of the infinite complexity of the world over finite combinatorics. Beneath the superficial similarity of events – and thus of description- lay a trove of infinite differences which superseded any finite number of combinations: paradoxically, ‘even if a previous century returns with respect to sensible things or which can be described by books, it will not return completely in all respects: since there will always be differences although imperceptible and such that could not be sufficiently described in any book however long it is.’. [Fi 72]’   Any repetition of event was thus only apparent; each part of matter contained the ‘world of an infinity of creatures’ which ensured that truths of fact ‘could be diversified to infinity’ (p. 77).

To this epistemological quandary Leibniz opposed a ‘cybernetic’ solution whereby the analysis of the infinite ‘detail’ of contingent reality would open up a field of constant epistemological renewal which lay beyond finite combinatorial language, raising the prospect of an ‘infinite progress in knowledge’ for those spirits ‘in search of truth.’ (p. 59) The finite number of truths expressible by humans at one particular moment in time would be continuously updated to adapt itself to the mutability and progress of the contingent world. ‘Sensible truths’ could ‘always supply new material and new items of knowledge, i.e. in theorems increasing in length’ in this manner permitting knowledge to approach reality asymptotically. In this manner, the theoretical limits which had been placed upon human knowledge could be indefinitely postponed, in the process allowing for incrementally greater understanding of nature through constant refinement.

Leibniz thus set forth an ingenious solution in the shape of a constantly updated finitude which would espouse the perpetually evolving infinity of concrete reality. By adopting what may be termed a ‘cybernetic’ solution avant la lettre, he offered a model, albeit linear and continuous, which could help reconcile determinism and probabilism, finite computation and infinite reality and freedom and predictability. Probabilism here served to induce and sustain a weak form of determinism, one which, in keeping with the nature of contingency itself as defined by Leibniz, ‘inclined’ rather than ‘necessitated’.

Audrey Borowski is a historian of ideas at the University of Oxford.