15.1 Introduction

Automation is all about representation and representation is always a political project. In order to hand off a given task to a computer, that task must first be reconceived and reformalized as something that a computer can do, translated into its languages, its formalisms, its operations, encoded in its memory.Footnote ¹ In service of those transformations, decisions have to be made about what is important, what will be lost in the translation, whose needs or goals will be prioritized. This chapter explores two influential attempts to automate and consolidate mathematics in the second half of the twentieth century – the QED system and the MACSYMA system – and the representational choices that constituted each: the languages of mathematics had to be translated into the languages and formalisms of computing; relatedly, mathematical procedures, like proof verification or algebraic simplification, had to be translated into computer-executable operations; and decisions had to be made about how best to formalize mathematics for automation, with what foundational logics, rules and premises.

MACSYMA and QED developers made very different representational choices and they used narratives to frame those choices. Marc Aidinoff has observed that historians often set out to unearth the ‘hidden politics’ of technological systems that are framed by their developers or users as value-neutral, objective, apolitical. He argues we should also ‘listen to people when they tell us what, and who, they prioritized’, we should attend to ‘the political, as it lies on the surface of technology, as actors directly described it’ (Reference Aidinoff, Abbate and DickAidinoff 2022). This chapter attempts to do just that by focusing on the narratives with which QED and MACSYMA were framed in order to make sense of the approaches to automation they represent, and the animating visions of mathematics and culture at work underneath.Footnote ² These narratives were not just stories, extraneous and external to the systems. Nor were they post hoc, developed to explain choices that had already been made. They mapped directly onto and informed technical development and design decisions. They also mapped onto practice – the representational choices framed by these narratives corresponded with cognitive realities – how users would have to think about and do mathematics with these systems.Footnote ³

As such, the narratives that framed each project were both political and epistemic.Footnote ⁴ They were foundational myths that advocated for the consolidation and automation of existing mathematical knowledge so that the computer could take over certain elements of mathematical labour – from algebraic simplification to proof checking – and in so doing open up new possibilities for knowledge-making. Mathematicians in the future, it was proposed, would be able to see new things, solve new problems and ask new questions with automated repositories of what was already known in hand.Footnote ⁵ Neither QED nor MACSYMA fulfilled their foundational myths, however. They were utopian narratives, at the intersection of political and epistemic imagination. Throughout the second half of the twentieth century, there was genuine uncertainty about what kind of tool the modern digital computer would turn out to be, what its epistemic and cultural limitations and possibilities were. The narratives explored here served to attribute meaning, possible futures and cultural values to mathematics as it would be made manifest in this new and undetermined technology.

15.2 Political Choices in Automation

The QED system, whose development began with an anonymously authored manifesto in 1994, was an attempt to combat the ‘tower of Babel’ its developers perceived in the automation of mathematics which had, throughout the 1970s and 1980s, involved a proliferation of ‘incompatible reasoning systems and symbolic computation systems’ that were inefficient, redundant, cacophonous, and that threatened mathematics’ traditional claim to universal truth (QED Manifesto 1994: 242). The QED Manifesto accordingly called for the translation of mathematics into a single formal and computational system, ‘that effectively represents all important mathematical knowledge and techniques’ and that conforms ‘to the highest standards of mathematical rigor, including the use of strict formality in the internal representation of knowledge and the use of mechanical methods to check proofs of the correctness of all entries in the system’ (QED Manifesto 1994: 238). It was to be a ‘monument’, gathering together, verifying and unifying mathematics, the ‘foremost creation of the human mind’. Writing in the wake of the Cold War, and amid the rise of American liberalism, the authors of the Manifesto proposed that the system would help ‘overcome the degenerative effects of cultural relativism and nihilism’ (QED Manifesto 1994: 239–240). They lamented the perceived loss of ‘fundamental values’ that the end of the Cold War and the rise of liberalism signalled and saw in mathematics a uniting and universalizing possibility.

QED would bring mathematics together by making it all the same – by formalizing it within one ‘root logic’, the same rules and foundations at work throughout. The Manifesto incorporated a narrative of ‘Babel’ and of the loss of shared cultural values in order to align their project with an ideological goal: they wanted to use the universality of mathematics in order to reinforce ‘fundamental values’, in the face of cultural difference. The home of the project was the Argonne National Laboratory (where some of the anonymous authors were based). This was an American government and military funded, Department of Energy hosted, effort to assert ‘universal truth’. But their project highlights that ‘the universality of mathematics’ is itself a construct. QED would make mathematics universal, by demanding that different visions, approaches, logics and techniques be put into one formal and technological system. Anything that wasn’t or couldn’t be reformalized in this way would be ‘outside of mathematics’, excluded from the centralized system, from the monument to truth. The corresponding commitment to shared fundamental cultural values is similarly normative – values will only be universal and shared when everyone has been convinced (or forced) to adopt them.

The authors of the Manifesto were right about Babel in mathematics automation. Since the early 1960s, there had been a proliferation of attempts to automate different parts of mathematics, and the resulting systems did not conform to shared formal or computational specifications. Some of the ‘cacophony’ resulted from the fact that system developers were building from scratch and without collaboration or communication with other system developers. Some differences were the result of direct competition between them. But some of the formal and representational pluralism was done by design, including in the second case to be explored in this chapter.

The MACSYMA system, developed at Massachusetts Institute of Technology (MIT) between the mid-1960s and the early 1980s during the Cold War, was among the most influential early computer algebra systems. It was designed with multiple representational schemes, multiple logics, on purpose, because the developers believed this would make it more useful to practising mathematicians and mathematical scientists. MACSYMA, too, was meant to be a centralized, consolidated, automated repository of existing mathematical techniques – a toolkit mathematicians could use in order to spare themselves the time and effort of learning and executing those techniques for themselves. But MACSYMA developers believed that the best way to automate and consolidate mathematical knowledge was with as much heterogeneity and flexibility as possible. They wanted to bring mathematics together in pieces, stand-alone modules that each operated according to its own logic, its own internal design. This, they believed, would create a more accurate and more useful encoding of mathematical knowledge that would reflect and respect the pluralism of mathematical communities.

In an article explaining the representational choices one must make in the automation of mathematics, MACSYMA developers used political language. In a section called ‘The Politics of Simplification’, Joel Moses (a lead MACSYMA developer) described these choices in terms of how much freedom they afford the user, acknowledging that user freedom almost always adversely affects efficiency (Reference MosesMoses 1971). There are many different but equivalent ways that mathematical relations can be expressed, and mathematicians choose particular expressions because they are convenient to work with in a given context. But what is convenient for a mathematician on paper may not be efficient on the computer where very different constraints and economies, of memory and operations, are at stake.

For example, even simple addition can lead to trouble on the computer. Consider the sum of a series of numbers [1] S = x₁ + … + x_n. In computers, numbers are typically stored in memory using a fixed number of bits, and for ‘real numbers’, a format called floating-point is used to represent them. However, floating-point schemes struggle to represent both very large and very small numbers. As such, for the purposes of automation when very large numbers may be involved, it might be simpler to work in ‘log space’ where the computer stores and operates on the logs of numbers rather than the numbers themselves, because they require less memory. **Incidentally, the capacity to simplify problems by calculating in ‘log space’ is what made tables of logarithms so valuable in the nineteenth century before automatic calculators.** Expression [2] $\log \left(\exp \left(x_1\right)+\exp \left(x_2\right)+\dots +\exp \left(x_n\right)\right)$ calculates the same value as [1], but works in log space, and so is often more efficient for computation. If you want to compute the log-space representation of the sum of x₁ to x_n, you can convert out of log space (by exponentiating), compute the sum of the regular representation of the numbers, and then take the log again, as in [2]. But, on the computer, it can be even more efficient to represent this expression as [3] $M+\log \left(\exp \left(x_1-M\right)+\dots +\exp \left(x_n-M\right)\right)where M=\text{max}\left(x_1, \dots , x_n\right)$ . [2] and [3] are equivalent, but how could [3] possibly be more efficient than [2]? It has this extra term, M, added and subtracted throughout. [3] is called the ‘log-sum-exp’ trick and it is a way of computing the sum of a series of numbers in log space without having gigantic intermediate calculations that could exhaust computer memory. While it complicates the expression by adding M, M simplifies the computation by ensuring that numbers are sufficiently small to be represented in available memory. But this way of looking at and working with sums may be counter-intuitive or difficult for a human user who may nonetheless be required to input expressions in this form or recognize and interpret them on the screen if sums have been implemented in this way in the system they are using. In this and so many other cases, what is easier and more efficient computationally may not be what is easiest for the mathematician.Footnote ⁶

Typically, the more representational flexibility a user has, the more ‘under the hood’ processing needs to be implemented by developers to translate inputs into a form that the system was set up to manipulate. A ‘user-friendly’ system might allow a user to input simple expressions like [1] and, ‘under the hood’, the computer could convert them into the more computationally efficient forms in [2] or [3] before executing, and then convert back when displaying a result. But, these conversions also cost computing resources, so more rigid designs demand that the user become accustomed to working with, recognizing and generating computer-oriented representations themselves. This problem – how to implement and represent mathematical expressions and operations efficiently in memory, how users could input and work with mathematical expressions and operations, and how much work was needed to translate between the two – is a core problem for the automation of mathematics. These are the representational choices involved in any automation effort, and these are the choices MACSYMA developers framed through political narrative.

Moses surveyed the algebraic computing systems of the 1960s according to what he figured as the politics of their representational choices. There were the so-called ‘radical systems’ that could only ‘handle a single, well-defined class of expressions. […] This means that the system stands ready to make a major change in the representation of an expression written by a user in order to get that expression into the internal canonical form’ (Reference MosesMoses 1971: 530). There was ‘the new left’, which ‘arose in response to some of the difficulties experienced with radical systems’ and which operated like a radical system but with some alternative algorithmic simplification mechanisms. There were ‘the liberals’, equipped with ‘very general representations of expressions’, the ‘conservatives’, who ‘claim that one cannot design simplification rules which will be best for all occasions. Therefore, conservative systems provide little automatic simplification capabilities. Rather, they provide machinery whereby a user can build his own simplifier and change it when necessary’ (Reference MosesMoses 1971: 532). There were also ‘catholic’ systems that used ‘more than one representation for expressions and have more than one approach to simplification. The catholic approach is that if one technique does not work, another might, and the user should be able to switch from one representation and its related simplification facilities to another with ease’ (Reference MosesMoses 1971: 532). MACSYMA was a catholic system, incorporating elements of liberal, radical and conservative representational choices – ‘The designers of catholic systems emphasize the ability to solve a wide range of problems. They would like to give a user the ease of working with a liberal system, the efficiency and power of a radical system, and the attention to context of a conservative system. The problem with a catholic system is its size’ (Reference MosesMoses 1971: 532). MACSYMA, with its catholic design, reflected a narrative that highlighted horizontal management – the system’s modules operated independently of one another – and pluralism – each module operated according to its own representational schemes and internal logic (Reference Martin and FatemanMartin and Fateman 1971).

Any attempt to encode and automate mathematics requires an answer to a host of representational questions – how should mathematical objects be stored in computer memory? What will be included and what will be excluded? How should human practice be translated into computer operations? Whose needs and perspectives will be prioritized – the user or the developer? How and how much should these processes and representations be made visible to the user on a screen or printout? How must users formulate their problems and objects of interest such that they can be input to the system? QED and MACSYMA were designed with different answers to this set of representational questions, both framed with politico-epistemic narratives. QED embodied a vision of mathematics as a source of universal, shared truth and ‘fundamental values’ in the face of scorned ‘cultural relativism’. MACSYMA instead embodied a commitment to pluralism and flexibility in both mathematics and culture. These narratives flag the cognitive freedom or discipline that accompanies different approaches to automation – they describe how users must discipline their relationship to mathematics and mathematical representation in order to use a system effectively. They imagine a different role for computers in the production of mathematical knowledge, and different ‘styles of reasoning’ to accompany them (Reference HackingHacking 1992).

15.3 From Political Choices to System Building

But how (and how well) do these narratives relate to on-the-ground realities of these projects? How free are the developers of technological systems to decide what their politics will be? What is highlighted and what is left out in these narratives? Jonnie Penn, a historian of artificial intelligence (AI) has demonstrated that, in spite of all of their self-proclaimed differences, early AI practitioners were in fact united by key underlying logics and values (Reference PennPenn 2020). While they disagreed about how intelligence might be manifested in the machine, or what intelligence was, different approaches to AI were nonetheless united by many shared commitments – most notably, he identifies military and industrial logics and funding at work across them. For all their purported differences, they in fact agreed as much as they disagreed, especially about unspoken assumptions. Similarly, on the face of it, QED and MACSYMA embodied opposite approaches to the same problem – both projects aimed to centralize and automate mathematics, MACSYMA by preserving difference and adopting representational flexibility, QED by translating all of mathematics into one ‘root logic’ by unifying it. The narratives adopted by the developers of each system correspond to these opposing visions of automation. However, in spite of those differences, both systems shared a more fundamental belief that the consolidation and automation of mathematics was possible. They shared an underlying goal – to extract mathematical knowledge from people and communities and put it into the machine. To do so, both projects had to accommodate computers, whose limitations and possibilities constrained the epistemological and political values they could realize. The next sections offer a closer look at each automated system, the narratives that surrounded them and the practices that accompanied them.

15.4 Conclusion

This vision – that mathematics will be fully consolidated, automated and formalized just around the next social or technical corner, that its universality will be made materially manifest – gained much traction in the late nineteenth and early twentieth centuries. Responding both to the discovery of several troubling paradoxes and to the proliferation of mathematical fields and centres of research, mathematicians around the turn of the twentieth century wanted to get all of mathematics into one place, they wanted to represent it all in the same formal system, the same symbolism and in the pages of one book. They were unable to do so, for formal, social and material reasons. With the perceived possibilities of modern digital computing in the 1960s and 1970s, many, including the developers of the MACSYMA system, believed that, finally, consolidation would be possible, especially through pluralism and horizontal management. It wasn’t. Again, in the 1990s, the anonymous authors of the QED Manifesto proposed that finally the cost of computing and the intellectual will were such that it would be possible to gather up all of mathematics in one place, in one formal system. It wasn’t. In revisiting the QED Manifesto two decades later, several mathematicians proposed that the time had finally come for the full and final consolidation of mathematics. It hadn’t. This story – that mathematics will be fully unified, consolidated, formalized just around the corner – now that the conditions of past failures have been overcome – shapes whole research projects, and scaffolds belief in the universalism of mathematics.Footnote ¹¹

In spite of their different approaches to automation, and the different narratives that accompanied them, QED and MACYSMA both participated in that shared goal of consolidating mathematical knowledge and automating it, putting it in the machine. Moreover, both received initial funding from the same organizations – DARPA and the Office of Naval Research (ONR), especially. Both projects were undertaken at powerful hubs of military–industrial–academic research, MIT and Argonne National Laboratory, whose power grew out of the post-war American context. Both ascribed to ideologies of efficiency and logics of industrial planning in their imagining of automated mathematics, but to serve two different ideologies. Both projects rested on the belief that, whether pluralistically or not, knowledge could be extracted from human knowers, that it could and should be ‘put into the machine’. And both set out to redefine, transform and encode mathematical knowledge with computer-oriented representations and processes. QED and MACSYMA have more in common than their framing narratives may suggest.

MACSYMA was meant to preserve pluralism and empower mathematicians for new programs of problem-solving. It was meant to free time and energy for new questions and explorations by handing over much mathematical labour to the machine. However, the freedom afforded by MACSYMA required users to work with and within highly disciplined and often counter-intuitive computer-oriented representational schemes, and that freedom cultivated dependency on the system, once a user came to rely on the system for the execution of techniques they did not themselves understand (Reference DickDick 2020). The developers conceded the point that MACSYMA required mathematicians to reconceive what they know for the purpose of automation, and even encouraged users to transform their own knowledge into automated modules for inclusion in the system. The modularity that was meant to serve a pluralistic and modular vision of mathematical practice also made it easier for mathematicians to take what they knew and ‘put it in the machine’. Users could contribute to a SHARE Directory – an ever-growing repository of new modules, user-generated, that expanded the system’s capabilities and made more ‘knowledge’ available to more people. The claim that MACSYMA freed mathematicians and that it preserved pluralism of practice betrayed the fact that incredible accommodation to the machine was first required and that the system was primarily useful and usable to elite and defence-funded institutions. When MACSYMA was privatized in 1981, and licensed to Symbolics Inc., the users who had worked so hard to learn and accommodate and even contribute to the system were then transformed into a set of buyers in a market who had to now pay for the privilege of consuming the goods they had in part made themselves. MACSYMA wasn’t the materialization of freedom and pluralism that its narrative suggests.

Lewis Mumford cautioned, in opposition to strong theories of social construction, that there are technological systems that cannot be aligned with any politics whatever, but rather operate according to fundamental logics that cannot be overcome through creative use, alternative intention or new narrative. Mumford suggested that computers are essentially authoritarian technics, centralized command and control technologies, no matter how often people have tried to align them with democracy, freedom, counter-culture and pluralism (Reference MumfordMumford 1964; Reference TurnerTurner 2008). Even if one doesn’t accept Mumford’s analysis in its entirety, it would still be safe to suggest that no American militarily funded effort to extract knowledge from knowers and communities and make it efficiently and automatically available to defence-funded research institutions, can be aligned with the politics of pluralism.

Both QED and MACSYMA were supposed to serve a dual purpose. First, both were meant to automate mathematics, and in this they differed – the former meant to automate by representing all of mathematics in a shared ‘root logic’, the latter, automating mathematics modularly, attempting to preserve logical and methodological pluralism, as well as offer users flexibility. In this difference, the narratives the developers attached to the projects suit. But both projects were also meant to consolidate all of mathematical knowledge, efficiently and automatedly. Both entailed and in fact celebrated the displacement of human understanding – users need not understand that which the system can do. For MACYSMA, users would be spared the need to learn mathematical techniques for themselves because of having an automated system available to execute them instead. In QED, the fundamentally social project of establishing mathematical truth was displaced in favour of automatically verified results. Both entailed theories of knowledge that did not require a subjective knower, only a machine encoding. And in this regard both displace human understanding, social processes and the pluralism these entail. And both projects consolidated resources and decision-making power, as well as the automated mathematical knowledge itself, in the hands of a small number of institutions, also limiting pluralism. Both projects also minimize the productive capacity of friction, miscommunication, disagreement, misunderstanding and difference. While MACSYMA preserved logical pluralism in its modularity, all modules still had to accommodate the constraints of a single arbiter: the PDP-10 computer on which they ran. We might call this computational-pluralism, and it was only as plural as those constraints permit. The politics of technology go beyond the technical design choices made within them to include the context in which they are developed, who pays for them, profits from them, and how much freedom or discipline users and contributors have in their engagement with technical systems.

In these histories of mathematics automation, narratives map onto design and implementation decisions, they acknowledge the representational choices involved in accommodating the machine and the user and they reflect beliefs about mathematics’ relationship to culture. But the narratives that developers use to frame their technological systems may also serve to direct our gaze away from certain institutional realities and unspoken assumptions. These epistemic–political narratives highlight entanglements between mathematics and culture, and conformity and freedom, in the representational choices that automation always involves.Footnote ¹²