Authors Meet Critics

Massimo Mazzotti, “Reactionary Mathematics: A Genealogy of Purity”

Recorded on October 17, 2023, this video features an “Authors Meet Critics” panel on the book Reactionary Mathematics: A Genealogy of Purity, by Massimo Mazzotti, Professor in the UC Berkeley Department of History and the Thomas M. Siebel Presidential Chair in the History of Science.

Professor Mazzotti was joined in conversation by Matthew L. Jones, the Smith Family Professor of History at Princeton University, and David Bates, Professor of Rhetoric at UC Berkeley. Thomas Laqueur, the Helen Fawcett Distinguished Professor Emeritus at UC Berkeley, moderated.

This event was co-sponsored by the Center for Science, Technology, Medicine, & Society and the UC Berkeley Department of History. The Social Science Matrix “Authors Meet Critics” book series features lively discussions about recently published books authored by social scientists at UC Berkeley. For each event, the author discusses the key arguments of their book with fellow scholars.

About the Book

A forgotten episode of mathematical resistance reveals the rise of modern mathematics and its cornerstone, mathematical purity, as political phenomena. The nineteenth century opened with a major shift in European mathematics, and in the Kingdom of Naples, this occurred earlier than elsewhere. Between 1790 and 1830 its leading scientific institutions rejected as untrustworthy the “very modern mathematics” of French analysis and in its place consolidated, legitimated, and put to work a different mathematical culture. The Neapolitan mathematical resistance was a complete reorientation of mathematical practice. Over the unrestricted manipulation and application of algebraic algorithms, Neapolitan mathematicians called for a return to Greek-style geometry and the preeminence of pure mathematics.

For all their apparent backwardness, Massimo Mazzotti explains, they were arguing for what would become crucial features of modern mathematics: its voluntary restriction through a new kind of rigor and discipline, and the complete disconnection of mathematical truth from the empirical world—in other words, its purity. The Neapolitans, Mazzotti argues, were reacting to the widespread use of mathematical analysis in social and political arguments: theirs was a reactionary mathematics that aimed to technically refute the revolutionary mathematics of the Jacobins. During the Restoration, the expert groups in the service of the modern administrative state reaffirmed the role of pure mathematics as the foundation of a newly rigorous mathematics, which was now conceived as a neutral tool for modernization. What Mazzotti’s penetrating history shows us in vivid detail is that producing mathematical knowledge was equally about producing certain forms of social, political, and economic order.

Listen to the panel as a podcast below.



Julia Sizek: Hello, everyone. Welcome. My name is Julia Sizek. And I am a postdoctoral scholar here at Social Science Matrix. Today, as you all know, we are here as part of our Author Meets Critics series in which we discuss exciting new works by faculty in the Social Sciences Division.

Our book today is Reactionary Mathematics– A Genealogy of Purity by Massimo Mazzotti. As promised by the title, the book is a look into the history of mathematics and more specifically the late 18th and early 19th century in the Neapolitan resistance to French styles of mathematical practice. This revolution in mathematics, Mazzotti argues, should be examined alongside the political movements at the time.

A pure mathematics, he suggests, is a project of a certain kind of political, social, and economic order. This wide-ranging book is of interest to many of you here. And perhaps you arrived thanks to the efforts of our co-sponsors for this event, the Center for Science, Technology, , Medicine, and Society and the History Department. So thanks to them.

Before we begin, I’m just going to discuss a couple of our upcoming events that we have coming at Matrix. So on October 31, Halloween, very spooky, we have the California Spotlight From Boom to Doom in San Francisco about the so-called doom loop, which I assure you is a very terrifying topic. On November 14, Dylan Penningroth will be presenting his book Before the Movement– The Hidden History of Black Civil Rights.

On November 28, Sharad Chari will be presenting his short book Gramsci at Sea. And then finally, for the purposes of this, we will be having a event featuring the work of graduate students called New Directions in Gender and Sexuality toward the end of the semester. So to register for all of these events, you just go to our website, which is

And so just so you know, the way that this event will proceed is first, we will have our moderator introduce everyone. And then we will proceed to have some discussion up here. And then we’ll open it up to Q&A around the room.

So I will be introducing our lovely moderator Tom Laqueur. Laqueur is the Helen Fawcett distinguished professor emeritus– wow, that’s a mouthful– at the University of California Berkeley. His work has been focused on the history of popular religion and literacy on the history of the body, alive and dead, and on the history of death and memory.

He writes regularly for the London Review of Books and The Threepenny Review among other journals and is a founding editor of Representations. Laqueur is a member of both the American Philosophical Society and the American Academy of Arts and Sciences. His most recent book is The Work of the Dead– A Cultural History of Mortal Remains. And he’s currently working on a book called The Dog’s Gaze in Western Art to be published very soon. Without further ado, I’ll turn it over to Tom.

Thomas Laqueur: Thank you. So we have a remarkable panel today to discuss this book, remarkable in the sense of the distinction of the panelists, but also that they actually know something about this subject, which is not always the case. So just to the left is David Bates, Professor of Rhetoric at Berkeley, but for many years very active in the Center for New Media and now in the Center for Science, Technology, Medicine, and Society.

He’s an enlightenment scholar. Has a book called Enlightenment Aberrations– Error and Revolution in France and States of War– Enlightenment Origins of Politics. He’s now working on a book on artificial history of natural intelligence.

And one further left is our visitor from afar, Matthew Jones. He’s the Smith Family Professor of History at Princeton. He focuses on the history of information technologies and artificial intelligence as well as the history of science and technology in early modern Europe. But actually, now, he’s not working in early modern Europe. He’s working on postmodern America, a history of surveillance. since 9/11.

But relevant to what we’re talking about today is how data happened, a history from the age of reason to the age of algorithms. And before that, a book on the history of science or generally The Good Life from the Scientific Revolution– Descartes, Pascal, Leibnitz, and the Cultivation of Virtue. And other books and works, but in any case also in this general field.

Massimo Mazzotti, our colleague and the main event, I want to introduce him actually by reading from the review of Nature. It’s a good place for– a rare place for a historian to have a– so here’s what Nature says. “There are some books that hook you– that hook you straight from the title. Reactionary Mathematics by Massimo Mazzotti is one of them. What’s the title even mean? It feels as a bizarre juxtaposition of two seemingly unrelated terms like literary biology or electrical jurisprudence.”

So then it goes on to say that, “Many people have perceived mathematics as separate words the most independent of these disciplines from the social science but not Mazzotti. Mazzotti’s first merit is to break this pattern and take us to a different sphere where mathematics, science, culture, art and society, and history converge, revealing new interpretive possibilities.”

“Indeed,” the review concludes, “the complex relationship between tradition and modernization is the pulsing heart of this engaging book besides a valuable historical analysis. Reactionary Mathematics offers an interesting and useful synthesis,” I had to correct his grammar, “useful synthetic vision,” he’s Italian, so you know. “to help us understand in these times of rapid and convulsive transformations, the mathematics of the present, and most importantly, the reason for the mathematics to come.

So what else can I say, except that Massimo is the Thomas Siebel Chair in the History of Science in our department. And his earlier book is also in some sense about the cultural history of mathematics, The World of Maria Gaetana Agnesi, Mathematician of God. So Massimo will speak. We’ll have both respondents. Matthew will respond. And then I’ll call on everyone for questions.

Massimo Mazzotti: Thank you. Thank you. Thank you, Tom. [INAUDIBLE] Thank you, Tom. So yeah, I think it’s probably useful if I don’t speak too much. And we have a conversation. So maybe I just give you some coordinates just to give a sense of where we are. Because one problem here is that it’s a fairly obscure story in the history of mathematics. So it’s pretty– I mean, darkness when it comes to my colleagues mostly when I talk to them about this.

So at the center of the book, there is a short chapter in which I talk about the mathematical controversy, which was not very relevant. I mean, actually, quite marginal to the overall history of mathematics in Europe in the 18th, 19th century. And that is actually the core of the story. But somehow, I didn’t know exactly what to put it. Because I really want to tell about things that have to do with that controversy.

So the book is mostly about the meaning of that controversy. And then at the center of the book– then I decided, OK, I put it at the center. It’s really at the center of the story. I’ll put it physically at the center as well.

So the story is relatively simple. It’s like, the controversy is about two groups at the margins of enlightened Europe, the Kingdom of Naples, which is well known but is well known for the Vesuvius, for some of the natural [INAUDIBLE]. So artistic natural historical remains. Not really a powerhouse in mathematics. That’s Paris. Those are some other places in Europe that are really– that kind of place.

But there is this controversy that goes on from roughly the 1790s to the 1830s, age of Revolution. And this has to do with what is the best way to solve one version of it. A simple version of it is what is the best way to solve a geometrical problem. Should we be using all kinds of algebraic techniques, even though they don’t really reflect the initial geometrical problem?

But it’s like, we turn the geometrical problem into analytical Cartesian geometry. But even more than that, you can actually use calculus and other things so that you really move distant from the original geometrical problem. You operate on those algebraic formulas. You get numbers. Those numbers, you interpret them as giving you the answer to the geometrical problem.

So you go back. And, OK, you solve the problem. Or should we stick with the geometrical– with geometry? And should we actually only reason in geometrical terms– which means like Euclidean geometry.

Obviously, we’re talking about fairly sophisticated versions of it. But still, should we actually be in the world of seeing geometrical figures either physically or without imagination or in the world of what they would call blind calculations? Because you don’t really see anything when you’re crunching numbers. You’re just crunching numbers.

So at the core is this kind of debate. And what is– I mean, this is just one of the many versions of algebra versus geometry, which is a long-lasting story in the history of mathematics. But at that moment, at that juncture of European history, it seems to me that it takes a particular significance. Also, because there is an unusual emotional investment into this debate, that it doesn’t immediately– it’s not immediately justified by the actual content.

And you find the supporters of the geometrical approach, the synthetic approach, synthetic geometry, meaning, essentially, Euclidean geometry, arguing that if you actually use the algebraic methods, you are perverting the mind of your students. Your mind is perverted. The results are going to be catastrophic for mathematics, which will be degraded, and for society at large.

Because you’re introducing a false certainty. You’re introducing false mathematical reasonings that then are used by people who, for example, are doing political economy or other things with your mathematical tools. And they trust you because as a mathematician, you have endorsed them. You have legitimated them.

So there is a question of what is the legitimated set of tools that one should be using. And this is invested immediately in epistemological moral terms and terms that often evoke a social crisis. This is guiding us into some kind of absolutely wrong direction.

So the trajectory of the controversy is really a kind of– as many controversies, it just disappears into insignificance at some point. It’s not relevant then in the ’30s. But between the ’90s and the ’20s, really, mathematical and scientific life– because this invest everyone who is using mathematics. So from the engineers to cartographers, anyone who’s actually a scientist at that point of natural philosopher in Naples is really one of the most heated and apparently significant controversies.

So what they do in the other chapters is to give you a sense– I zoom into the controversy from the point of view of the analytics, let’s call it that way, those who are arguing for the value of algebraic reasoning and the power of algebraic reasoning so that we see the story from their side. The story from their side is a story of the trust in universal analytic reason, let’s call it that way, that is essentially reflects the deep structure of nature. And it also expresses the deep structure of our own mind. The two things are isomorphic.

And you might think of French mathematics in the late 18th century, the likes of Condorcet and others are really thinking along those lines. I mean, this is not just a mathematical technique like others. This is what they call analysis, which is now what we call analysis today is a set of techniques that have to do with algebra, calculus. But it’s not coherent theory by any means.

But what do you do when you’re using those techniques, you’re actually using something that is deeply ingrained in the human mind and in nature. And so the legitimation of those techniques is the world somehow. It doesn’t need to be grounded into something else. I mean, that’s not a problem that they are thinking about.

Historically, the way in which I see this coming together into the controversy is that there is a phase in which these kind of arguments are used by reformists in late 18th century, Naples to argue for essentially criticizing established institutions, social institutions, and often suggesting some transformation. So it’s a kind of a reformist push based on mathematical arguments.

So there are better ways of, for example, organizing a certain productive process. The rules of algebra are telling us– are telling us what these ways are. So they’re guiding us. They can guide us rationally. Things get a bit more extreme in the ’90s when, essentially, after the Bastille and the 1789, the government, the king and the court turn actually dramatically on a kind of anti-French and conservative side.

And at that point, any argument that might sound like universal reason applied to the transformation of society is not really very welcome anymore. And, in fact, people who have been marginal in the scientific world up to that point, an interesting group of mystically inclined mathematicians who were defending some essentially invented tradition of Euclidean geometry become central to scientific life in the Kingdom of Naples. They become professors at universities.

I mean, really they occupy all the possible spaces in that world. So in the term of three years, you see a completely different scientific world in which now, the essentially scientific life is controlled by those who believe that Euclidean geometry should be the only basis of any mathematical procedure.

So just to cut it short in the relevance of this in terms of why do they care so much, well, if you argue that there is an unrestricted possibility of applying algebra and calculus to reform the economy, political life, creating an electoral system the way that Condorcet does, for example, trade, the landscape– so civil engineering. Civil engineers are legitimated by their own use of analysis in reshaping, for example, designing new roads; unifying the system, standardizing weights and measures. So that is what is being legitimated by the kind of universal rationality. Let’s say that the idea of a analytic reason that goes beyond any contingent expression of technical and mathematical knowledge.

On the other hand, the move to Euclidean geometry as the only legitimate foundation is a way of restricting the use of mathematics in a particular way so that you– mathematics is still core in the university curriculum the way that it was. It was a significant discipline. But now, it’s the movement of techniques from mathematics to other fields that is much more complicated and not legitimated. Because geometry– I mean, as Galileo realized, you can do only so much in terms of quantifying reality and transforming reality through quantification with Euclidean geometry.

So this was the– the Neapolitan story is the story of essentially restricting the possibility of using certain algebraic techniques. Because these algebraic techniques had been used by the Jacobins, who, essentially, in 1799 are able to seize power just for five months and set up a Jacobin Republic. And if you look at how they organize the republic, is kind of an analytic republic.

So they have a way of thinking that is pretty much the way of thinking of the analytical mathematician. And, in fact, it’s interesting that the president of the republic and one of his main assistants, first thing they do, they publish a textbook of mathematics. Because they say, well, it is this the way we need to think about the world.

Because if you want to transform the world, you need to analyze it, which essentially means to break it down to its elementary components– this is one of the old meanings of analysis is this– and then combining these components– so the combinatorial element of analysis– in order to construct something new based on those components. And how do you construct something new? Following the universal rules of algebra, which somehow is giving us– is guiding us.

So once the Jacobins have essentially built up their own discourse around– I mean, egalitarianism, anticlericalism, redistribution of wealth analysis, I mean, that’s what you find in Jacobin texts. At that point, analysis– I mean, it’s like, you cannot go back to analysis without being associated with the Jacobin– with Jacobin politics.

And so the reactionary mathematics of the title is literally the reaction to that moment. It’s literally a moving mathematics away from a conscious self-reflective way of using mathematics as a transformative tool for redistributing agency, essentially, across society, because that is at the bottom. That’s what they were doing– redistributing political agency across society in a way that would empower subaltern groups that had never been empowered.

The reaction is to make that a logical impossibility, a mathematical impossibility. And you make that by saying that that mathematics is not reliable. Obviously, we’re not just talking about Euclidean geometry. And the example I can give is that this is actually a much bigger story than Naples.

And if after having– the Neapolitan story is instructive because it’s so extreme, that you see everything is in your face. The political value of mathematics is there. They talk about it. They write about it. Whereas if you look at Paris, for example, the main place at the time where Augustin Cauchy is revolutionizing mathematics, as many historians have said at this point.

And what is he doing? It’s not going back to geometry. That’s a bit of a bizarre idea that could only happen in a marginal place is restructuring algebraic and essentially calculus, algebraic techniques and calculus. The term is rigorization, the rigorization of calculus. You may have heard that. This is something that happens in the first half of the 19th century.

And essentially, the outcome is pretty much modern mathematics, as we know it. And this rigorization of calculus is a way of restricting the use of calculus. It’s a way of saying, well, you need to be really precise enough of this voluntaristic, enthusiastic 18th century d’Alembert-like use of mathematics.

Tell me exactly what you’re doing. And tell me exactly what are the limits. There are many new things that comes up around this time, which are all designed to specify under which conditions certain formulas can be used and for which quantities of the values that are part of the formula. Because you cannot just give for granted that you can use any formulas, apply to any field, and without restrictions.

So if you read that after having come across the Neapolitan story, you see that what he’s doing is the same thing that this bizarre Neapolitan mystic mathematicians were doing in a more bizarre way, I would say. But it’s the same thing. It’s restricting the applicability of mathematics, particularly to issues that have to do with politics and theology and metaphysics. He’s arguing that there are many kinds of truths, that not everything is reducible at the same level of– at the same epistemological level, and that the mathematicians, they talk about the world of pure mathematics.

That’s the purity of the title. Pure mathematics at this point is becoming the foundation of mathematics. And why do we need the foundation of mathematics? Well, because if mathematics is not embedded in the world anymore or in our own reason, then we really need something like a foundation for this body of knowledge.

And the foundational crisis– and we have a few of them in the history of mathematics, but this is one. And the anxiety that it provokes, the fact that people like the Neapolitans or Cauchy are really anxious about the scandal of the lack of foundation of mathematics is, in fact, the scandal of the unrestricted application of mathematics. And the fact that we need to ground it, it needs to be understood as a self-included body of knowledge that is really not the essential structure of reality, something else.

So then you have all these considerations about how come the mathematics is effective, for example, if it’s– because if you start to think of it as something completely different, detached, and somehow endowed of a purity– purity meaning is not polluted by empirical considerations, which is a very modern way of thinking about mathematics. Because if you go back in time, most people think about mixed mathematics.

Mathematics is always something in between. It’s like astronomy or music, whereas this is like a distinctively modern way of thinking of mathematics as something that is really its own world. So then it takes a lot of work to make it function. And you need to justify any use of those techniques into the real world.

So I think that’s probably more than enough. But just to give you a sense. That’s what essentially the story is about with a lot of detail.


Thomas Laqueur: So do we have an order– do we– David?

David Bates: I think I was next, I guess. It’s hard to go after the star of the show. I’ll keep this kind of short. I have lots of notes. But just picking up on your last point, I think it’s more than just that story and some details. The book is actually much more unruly than Massimo gives it credit for.

And I mean that in a positive way. But I’ll start by saying a brief anecdote. When I was a graduate student at the University of Chicago, my first semester, I took a course from the cultural historian Carl Weintraub. I don’t know if anyone reads him anymore.

And for some reason, I got tagged as the postmodern kind of Yahoo of the class, constructivist, relativist, and basically willing to undermine all these kinds of values. But there was one point in the class when I guess I’d read Kuhn and Feyerabend when I was an undergraduate. So these things just came kind of naturally.

But at one point, he said, but what about mathematics? And he kind of stopped me in my tracks because I couldn’t really– I didn’t have anything to say about that. How do you historicize something as pure as mathematics? So that’s sort of stuck with me as I read more in history of science and history of mathematics.

But I never really read something as good as your book, to be honest, that really took that seriously to demonstrate how the most pure, logical step has to be understood, contextualized in a very rich and detailed way. So I’ll just admire and suggest that you read the introduction, which really gives a really good overview of some of the ways that mathematicians approach history, both for better or for worse.

But I think what’s really brilliant about the argument here, as Massimo’s kind of demonstrated already, is that at its heart is an idea that the very idea of the logical purity or the neutrality of mathematics has a history and that we have to do this kind of very close work to understand what’s going on in that longer history of mathematics.

So in the introduction, we get this really interesting view, which is that we don’t have mathematics so much as a mathematical culture. And that culture includes a really important– what he calls an image of reason, a kind of practice, as well as theory of reason. And this image of reason is predicated on concepts of order. And then we can slide nicely into the repercussions, which is essentially that any mathematical culture is going to have some implication on the social and political plane because concepts of order and concepts of rationality infuse what we really mean by social and political action.

So the introduction takes on that job of showing exactly what a history of– I even want to say it’s maybe not a history of mathematics, so much as a history of reason with a mathematical core. But you really do have a larger scope in the whole book. It’s not just about the specifics of the mathematicians but the image of reason and the image of order that goes along with that.

So now, I have armature for that. If I could go back in time, I’d have armature for how to demonstrate– you have this lovely line that what counts as a step in logical deduction always had to be constrained by this mathematical culture that every step in a logical deduction has to be understood in terms of its context. OK. So the book, as you described, is like an X. It has this core, which is probably the most mathematical part of the book.

It’s kind of like a textbook. It really teaches you what the difference between analytic and geometrical forms are. But it also raises these bigger questions. Basically, the synthetics are arguing, as Massimo says, that the art of inventing is not algorithmic and can never be algorithmic, whereas the algebraists were arguing that analysis could be understood as a universal form of reasoning. Those are Massimo’s phrases.

But analysis is the kind of catching point of the whole book. Because it can be celebrated as well as denigrated as mechanical and automatic. I think that’s one of the interesting threads of the book, the fact that analysis has, in some ways, an agreement on both sides that it’s mechanical character is really essential to understanding it.

So the book is an X with this sort of central core. The first part of the book follows the algebraic world, and the second part of the book is the synthetic world that I really do want to say that I think that you’re underselling the book by giving it this narrative of mathematics. Because what really happens is, like I said, it’s kind of kaleidoscopic and at times unruly. It delves into all sorts of different topics. I just realized the Library of Congress also undersells your book. It says it’s the topics are mathematics, study and teaching, Italy, Naples. Mathematics, political aspects.


When you read it, the first four chapters are actually– if Naples was a marginal space in this period, it was a pretty entertaining one. So what you have is really a political history wrapped up in a history of science, wrapped up in a history of mathematics, wrapped up in a history of administration. It’s really rich territory.

And it flows between different kinds of characters and different zones. But the intricacies of the French incursion and the reaction and then the French decade, as it’s called, is really interesting stuff. And it really follows, I would say, these mathematical cultures and even larger cultures of reason as they battle out in a number of different spheres.

So just for example, chapter 4 on The Shape of the Kingdom, it’s really like many of these great books that you’re citing that study the Enlightenment and 19th century that gets into the conceptual world behind everything from infrastructure, reform landscape and cartography, and excursus in the history of statistics in Italy and its use in political economy, the attempted transformation of the weights and measures system.

These are all well beyond just the mathematical debate that you take as the core of the book. And it’s really interesting stuff that pays off in a beautiful way with an analysis of landscape painting as well. So this is the kind of book. You could show some of those pictures in there after if you wanted to.

The last part of the book, I’ll just say, they’re not quite the heroes. But these sort of weird mystic synthetics are treated really generously, is the word I think, by Massimo. It takes seriously their concerns and I think makes the argument that I’ve also made a similar argument with respect to conservative thinking, that there’s no such thing as a going back, that these conservatives are really reactionary, and that they’re forging new models of politics, of science or reason. And you take that seriously.

So I really appreciated the last four chapters. Again, quite kaleidoscopic. We have discussions of Neo-Catholicism, of Demeestere and Bonald on questions concerning history and sovereignty. We have a number of fascinating mathematical tales all kind of interwoven.

And I think it really plays out beautifully the last half of the book. So again, I recommend this book for anyone interested in thinking about the role of science in policy, politics, but more, I would say, at the heart, this concept of a kind of culture of reason, a culture of order, how social political questions just are endemic to that space. You show that over and over again really, really brilliantly. So I’ll end with just a couple of questions.

In the last part of the book, the last chapter, you repeat this claim, which is to say that mathematics and especially the question of mathematical purity kind of occupies this essentially political space. And one thing that struck me just thinking about the book having read it, is to what degree this was an opportune moment to show, is this just the case of the always political aspect of mathematical practice or is this a special moment in the history of mathematics, and science, and politics that opened up a particularly rich opportunity to bring together Jacobin politics, or reactionary politics, or Catholic politics reform movements and so on?

To what degree do you take that any mathematical kind of culture is inherently political? And I ask that partly because one of the implications to me in the book, and again it’s because you’re so generous with the critics of algorithmic, algebraic mathematics, is to what extent the origin of our own algorithmic culture can be found in this particular period, the victory, let’s say, of algebraic analysis.

To what extent is the book? And you sort of– you hesitate to talk about this. And maybe I’m pushing you too far. But to what extent are you preparing the ground for a call for a new reactionary mathematics? Which would not look at all like the Neapolitan one but might have some kind of interesting resonance with contemporary ideas that go outside of this idea of calculation and prediction. So I’ll end there. But thanks again for the opportunity to read. It was really, really fun.

Thomas Laqueur: Great. Thank you. Yeah. Matthew? Do you want to respond? No? No. Matthew. Right.

Matthew L. Jones: OK. Well, thank you, Massimo, for the wonderful book. And David frames this beautifully. And indeed, I think much of what I’m going to say is thinking through how the unruly dense contextualization that Massimo provides gives the book so much of its place as a lever in thinking through major questions, I think, of both historical practice, science studies, and indeed the kind of social theory, which I take to often happen here in the matrix, that it’s very much a book about STS, Science and Technology Studies, and history and tandem, and in tension, and tugging on both, particularly by the case of mathematics.

So David mentioned mathematics being the hard case. And some of you will know that David Bloor, one of the founders of the sociology of scientific knowledge in a book called knowledge and social imagery, at first, articulated the symmetry principle. And in the symmetry principle, you treat that, which we hold to be true and that which we hold to be false symmetrically for explanation.

And he said, the hard case is, of course, mathematics, and I’m going to do that. And remarkably, not very many people followed him in doing this a few, including Donald Mackenzie, and Massimo, and several other of us have been profoundly fascinated. But there’s something about mathematics being challenging that sets it to be a challenging target for social theory.

But also, I think provides it with the power to thus serve in the kind of critical ways that I think that I intimate, that I think Massimo you’re getting at. And most of my comments are going to be about what I take to be, maybe I’m reading too much, many of the implications of what this local study helps us see, how it dislodges, how it helps us see particularly, precisely a reactionary mathematics, which are the nature article.

That weirdness makes it– allows it to serve as a lever for further analysis rather than a substitute for a deplored rationalistic present. It’s a way of looking at the past, where nostalgia is the subject of the analysis rather than something we fall into. So reactionary math is a powerful tool. And many of you may have encountered the way that say Edmund Burke is often lionized in odd ways today for his organicism, his anti-imperialism, and his anti– and it’s a peculiar move that leads to a kind of nostalgia.

And you quote Mark Lilla, who says that the reaction to the French Revolution has placed a cloud over European thinking. And it seems to me much of this book uses reactionary mathematics to help us think differently through this, reactionary mathematics from the periphery. So the question I really have is about, how does your book help us see the historization of different kinds of mathematics?

Mathematics in the plural. Try to get Microsoft Word to let you write logics in the plural. It tells you it’s not something you can do. Mathematics is in the plural, so it lets you do that. And that– so it’s like literally hard wired into ours. But how is different, looking at multiple mathematics, a lever against facile narratives and teleology, say of rigor but also nostalgia of a pre-quantified society, a pre-mathematical– how do we use that without falling into them?

So I’ve already spoken too much, but I’m briefly going to talk about nostalgia and the romance of the non-quantified and nostalgia and the romance of the localized past. So what I mean by the romance of the non-quantified, it’s enormously common across a whole wide variety of historical thought, social theorizing, and I don’t know, folks epistemologies of the dangers into which we’ve been thrust by quantification, much of which I share.

But your book, by reminding us of the non-uniformity of mathematics, pushes against too facile a narrative of what it means to become mathematical or quantified, by distinguishing between the analytic and the synthetic, and then in part way through, the morphing of the analytic into the technical, the statistical, the cost accounting. It asks for us to recognize that plurality.

And then to ask what are the plurality of purifications to be explained, we haven’t really talked about this so much. But one of the things that Massimo shows beautifully is a kind of purification that happens in synthetic geometry in which metaphysics and theological truths are outside of the domain of mathematics on the one hand. And another one, in analysis in which that hubristic ambition to transform everything along egalitarian lines is tamed, and it becomes a merely technical discipline.

So your work asks us to specify what those purifications are. And both in content and in cause. And to think, therefore, differently about very big stories that we often think about in the history of science, questions of how is it that we become disinterested or bracketed from, say metaphysical concerns or political concerns? And you do have two very different stories of that.

Mathematics is independent of and beneath metaphysics, not the Kantian story but this counterrevolutionary one. And then a neutrality of a counterrevolutionary statistics. The transformation of Jacobin math into a liberal sorts of things. A transformation of it into a technical, analytical quantification as a potential master discipline.

Now, I take it here. I’m reading into the text. But you are thus have a local history that speaks to our very current concerns about quantification and its claims to mastery. And your reactionary critics is often were right about seeing– they were right that the reformation was a big part of the problem because it had the wrong vision of the social organization of who was allowed to know.

So Descartes was accused of being an enthusiast. Because like the Protestants, he thought everyone ought to be able to opine. But here’s a question I was wondering for you. In your account of analytic mathematics, you talk about it being limited into a technical– in a way. But it’s not quite the story of Koshi.

Because it’s not that it can’t be applied to all domains of society. It is the case that it’s not applied towards illegality egalitarian means. And so I feel like your story is often very symmetrical. But you do treat in more fulsome detail the counter revolutionary mathematics far more than the analytical ones.

And so is it not the case that the purified analytic mathematics is just as hubristic and revolutionary as the Jacobin one. But it’s anything but egalitarian. It is hubristic, all expansive. But the shift is less than the structures– it doesn’t ask for a shift in the structures of society so much as a limitation of egalitarian hubris. OK.

So all of this is to say that the book pushes against a romance of a non-quantified society by showing the plurality of possible quantified societies. And the second kind of nostalgia I want to talk about is the romance of the localized path. So you begin the book by reminding us of symmetries and symmetric explanation, and again to return to the original progenitors of the sociology of scientific knowledge, a kind of synchronic symmetry.

But it seems to me, and maybe I’m wrong about this. But one of the most interesting undercurrents of your book is a kind of diachronic symmetry. And you’ve show so clearly how the reactionary account is a nostalgic vision of the past, not something we ought to return to but something that is quite constituted. David underscored this as well.

And I wonder to what extent you are there for pushing against certain kinds of history, histories of technology, histories of labor, certain kinds of social history that for all of their analytical fortitude of the dangers of certain technologies often resolve into a nostalgia of a world we have lost. And that can be analytically powerful as in the Jacobin coolest book Men and Measures, which pushes us against the teleological history of the metric system.

But it can also be limiting by immersing us in a nostalgia, which is a construction of reaction. How do we do dense local history of controversy but resist the pull of those modern alternatives? So my question is that one of the goals of the book to navigate thinking of that. And then finally, just one sort of question.

Your book is everywhere thinking from the periphery. And once or twice, you mentioned texts like provincializing Europe or provincializing the Enlightenment. But I wonder if you might be a little more explicit about how thinking from the periphery enables at once– as it were, a non-nostalgic history of other pasts and other sorts of presents.

And above all, how is it that thinking through multiple mathematics helps us push against the romanticization of the past we have lost and then think differently about debates I know you’ve thought a lot about, about the current nature of quantification and dangers of rationalization, often, which gets subsumed into some sort of an analytical amorphous mass of alienation from ourselves because of the mathematical? If mathematical is plural, then that obviously can’t be uniform. So thank you very much for your book.

Tom Laqueur: Thank you. Massimo, would you like to respond briefly?

Massimo Mazzotti: Yeah, briefly. I’ll try to get some of the things. Yes. I mean– OK, let me start from this. There are these two ideal types that I use, the analytic reason and the reactionary reason. Because in a way, as it has been said, this is a book about reason and the history of reason.

I mean, mathematics is the rationalizing practices that we often use to map reason, what kind of reason is in action here. And you see that through mathematical practices. That’s at least a good place to explore, to understand what kind of reason are these people giving for granted, what the idea of reason.

So obviously, these are two ideal types. And I mean, the story is about something that happens in between. There is always like– it’s a spectrum of different position. And in fact, I think what is interesting is often what is happening in between, which goes back also to the point of the liberal and the neutrality.

Because somehow, I often talk about the extreme version of the Jacobian analytic reason and the revolutionary emancipatory use of mathematics that they make. Then there is the reactionary, extreme version that I describe as having its own mathematics and its own set of cultural formations.

But at some point, in what we call the age of restoration, what we see is that really what becomes mainstream is neither of those. The reactionary option politically is dead by the 20s. I mean, no one really thinks that there is going to be any kind of return to the pre-revolutionary world.

And the Revolutionary option is kind of survives, underground, and in many different ways, but it’s definitely not on the table. So what’s on the table is using the case of Naples, but also elsewhere, is this kind of the liberal option. And the liberal option is one that interestingly takes on some of the elements of the analytic tradition but detached it from their own revolutionary potential.

And so the creation of the technical, as a space that now is the space of the engineer, the statistician, the cartographer is essentially– Benjamin Constant is talking about the emergence of this new space in very interesting political terms. But it is also a technical transformation.

Because now, there is this space that is the space for the technical elites that work for the government, in the continental context at least, that are using this new kind of is still analysis. So on the surface, there is a lot of continuity. It’s not that– and that’s a tricky thing about mathematics, right.

A lot of the techniques are the same. But now, the meaning of those techniques and the scope of those techniques is different. So to me, that’s one of the most interesting– it was one of the most interesting things that I came up with. To see that that’s also where the genealogy of our own world can be traced.

Because the rest seems really quite distant in many ways and eccentric. But if you look at that moment of the creation of this space, the technical space as a neutral space. Because that’s also– that’s a new thing. Jacobins would never think that the analytic tools they use are neutral.

They know they’re not. And that’s why they value them. They are programmatically impure mathematical tools. On the opposite, we have the kind of reactionary take. And what you have in this middle ground is a neutral technocratic often takes this kind of technocratic aspect is the courts of the civil engineers.

On the French example, you have this kind of courts all over continental Europe. The knowledge that they use is highly considered– is a powerful kind of knowledge, is a neutral knowledge, is powerful because it’s neutral because it doesn’t side with anyone politically. That’s why the engineers are powerful.

Because what they say is that’s the voice of science. That is the result of a neutral calculation. The fact that we need to build a road this way rather than that way, that’s what engineers are saying. So one of the ways in which I can see connections with the reflection that we often make about the present moment is in a way is it is a story of the giving for granted, the neutrality of certain technical tools.

And that is just something that emerges at a very specific moment. Because neither the revolutionaries nor the conservatives think that their mathematical tools are neutral. The conservative– the reactionaries are very aware that the way they do mathematics has an impact on the rest of society.

So they willingly restrict themselves to a certain kind of use of mathematical tools. So at that moment, let’s say the ’90s, in the midst of revolutionary action, I mean, no one is arguing the math is actually neutral. But that’s the outcome if you want to pin it down of the Napoleonic normalization.

I mean, that’s something that can older why it shows very well with the techno Jacobins that become the new elite of the engineers. So in a way, I was following this from the mathematical side. So that’s one way and part of the story. I’m thinking about something else that David was saying.

Any math culture is political. I mean, is this something that I would say? I would say this is an interesting moment because the imagination of these people is overwhelmingly political. I mean, this is a moment of unprecedented crisis, or at least they think it is unprecedented crisis. And so all they think is social order.

I mean, either for restoring it or for transforming it. So the fact– so the mathematical imagination is not detached from this overwhelming set of concerns. So by that standard, I wouldn’t say that necessarily at any time, when we consider mathematical cultures, there might be different.

A concern for social order is necessarily the first thing that you would immediately notice. But it’s definitely the case at this point. And also, more generally, something that I think is actually always the case is that when we construct structures, like logical mathematical structures like all this one, we create new techniques, new mathematical techniques.

What we are doing is we are creating new possibilities, new possibilities of thinking about the world in different ways, organizing the world in different ways, ordering the world and reordering it. So the more techniques we have, the more we can think that we can reorganize what we know in different ways that are legitimated by mathematical logical structures.

So if you think about this, then reducing mathematical– the possibility of certain mathematical options means reducing– restricting our political imagination. We cannot imagine at that point, or at least it seems illogical to think that the world can be very much different from the way it is because we have restricted the possibility of imagining structures that are completely different.

So this is one way of reading what I was saying before about the restriction of the legitimate mathematical techniques that you can deploy in, say thinking about political order. So this, I think, is a constant. The fact that depending on what mathematics you have, you will tend to think about different possibilities in ordering and reordering the world.

The fact that this is necessarily the first, most obvious, and overwhelming social dimension that is expressed by mathematical techniques, that is more contingent, I think. There might be other priorities and other things. And– sorry.

Tom Laqueur: No, go ahead. We should open the floor, but we can also– you guys can come back–

Massimo Mazzotti: On the thing, yes. I was really– that was really something I really care about. And I think that’s an important point. And I hope, yeah, my sense was to– in a way, the idea of mathematical cultures as an antidote to that sort of modern, pre-modern quantification, post quantification. That’s not a real divide.

Laqueur: So the floor is open.


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