# Review of Reactionary Mathematics: A Genealogy of Purity by Massimo Mazzotti

Andrew Arana

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## To cite this version:

Andrew Arana. Review of Reactionary Mathematics: A Genealogy of Purity by Massimo Mazzotti. Physis; rivista internazionale di storia della scienza, 2024, LIX (2), pp.614-618. 10.1400/298761. hal-05137675

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Submitted on 1 Jul 2025

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MASSIMO MAZZOTTI, 2023. *Reactionary Mathematics. A Genealogy of Purity*, Chicago
University Press, 352 pp.

The book under review is a microhistory of a clash in Naples around the turn
of the 19th century over the best way to do mathematics, especially geometry.
The book's title, *Reactionary Mathematics*, marks the political context of this clash.
With a short-lived Jacobin revolution in Naples in the 1790s and a decade of oc-
cupation by the French under Napoleon came what Mazzotti calls a revolution-
ary mathematics, analysis in the style of Lagrange. This revolution engendered a
reaction calling for a return to synthetic methods, a reaction that also expressed
itself politically against French occupation and that found its denouement in the
Restoration of the monarchy of the Kingdom of Naples.

That this is a microhistory means the book is full of local detail. Central fig-
ures, like the engineer Carlo Afan de Rivera and the mathematician Nicola Fer-
gola, are unknown outside this Neapolitan context. This aligns with the book's
framing of analysis as a universal method for acquiring knowledge and on syn-
thesis as a local method. In attending so carefully to local specifics, we are drawn
to sympathy for the reactionaries, who see the hegemony of French analysis as
part of a wider threat to Neapolitan ways of life. As readers today we have long
accepted the centrality of analytic methods, and may find ourselves scoffing at at-
tempts to slow down mathematical progress by insisting on synthesis in the style
of the ancients. The glorious chapter 4, on the shape of the Kingdom of Naples,
shows how there was something beautiful in the old ways that would be lost with
modernization – even as we recognize how these old ways were part of a feudal
society that left most poor and powerless.

Moving back and forth between mathematics and politics is unfamiliar, and
one may wonder if such an approach threatens the autonomy of mathematics. To
this, Mazzotti replies:

Recognizing the political dimension of certain mathematical choices does not mean that these choices were not truly mathematical: they were mathematical and political at once. After all, in Naples, French analysis had shaped the political imagination of local Jacobians and had been defended by French guns, deployed by civil engineers to reshape the landscape of the country, resisted by local communities, and mathematically refuted by reactionary elites. How could one split the two stories apart without missing the point of it all? (p. 9)

Readers who find this risible will do best to stay away from this book. For
those of us who agree wholeheartedly (see Arana and Burnett 2023 for my take
on this), this book will serve as a rallying cry.

The book's subtitle is *A Genealogy of Purity*. Mazzotti takes the term “gene-
alogy” from Nietzsche and then Foucault's application of the idea to the social
sciences. Alasdair Macintyre has characterized the goal of genealogy in this sense
as “to write the history of those social and psychological formations in which the
will to power is distorted into and concealed by the will to truth” (cf. MacIntyre
1990, p. 39). Nietzsche's idea is that what has been called "truth," that which can be seen to hold "from an objective point of view," is a construction that serves the peculiar interests of groups of actors like religious people, philosophers, and scientists. Rather than recognizing what holds from particular points of view, what Nietzsche calls "perspectives," these groups take their particular point of view to be a "God's-eye view," or "a view from nowhere." This creative act, behind both science and religion, can be seen, from the genealogical point of view, as an act of power. We can then look at how this power came to be realized, and at its consequences. For his part, Mazzotti explains that "this genealogy tries to reconstruct the concrete ways in which socioeconomic formations created the conditions of possibility within which certain mathematical objects – including the modern concept of “pure mathematics” – were constituted and legitimated." (p. 10)

Turning then to the second keyword of the subtitle, “purity:” there is no precise definition of this term in the book. The only quotation from an actor using the term occurs on the last page of the book. Nevertheless, the term is used enough that we can infer from context what Mazzotti intends. Mazzotti’s choice of this term seems to come from the distinction between ‘pure’ versus ‘mixed,’ ‘empirical,’ or ‘applied’ mathematics. As he writes on p. 227:

> Up until the eighteenth century, pure mathematics was understood as the result of a process of abstraction from empirical reality, with “mixed mathematics” occupying the intermediate space. Pure mathematics as it emerges after 1800 is something different. It is the very core of mathematics, and its separation from empirical reality is not one of degree: it describes another kind of reality – one that is purely intellectual, and independent from any empirical state of affairs.

Earlier, on p. 10, he distinguishes “the alleged purity of mathematical knowledge – as opposed to the impurity of knowledge derived from, and therefore polluted by, empirical considerations.” Purity in this sense is thus an absence of empirical content. This absence can be expressed in a more positive way. On p. 2, he writes:

> Through the prism of this controversy, I recount the dawn of mathematical modernity as a process of purification, a process whose outcome was the distinctively modernist perception that mathematical knowledge – and the technologies it legitimates – are neutral tools that can be used unthinkingly in the manipulation of both natural and social realities. This is a book about the genealogy of this image of purity-neutrality and its political meaning.

Purity for Mazzotti thus means neutrality with respect to subject-matter. Such neutrality is, he thinks, exemplified by analysis, where the particulars of a geometric problem – the curvature of a given ellipse, for instance – do not guide the solution. The analyst instead applies a general method of algebraic translation using coordinates, then calculates algebraically and numerically without thinking of the geometrical object in question, and finally retranslates the results of this calculation into terms that can be seen as having solved the problem. Analysts of-
ten described this method as mechanical, lauding its uniformity for problems that varied widely in their particulars.

The Neapolitan school of synthetic geometry, led by the aforementioned Fermola, reacted against analytic geometry. They praised synthesis for its rigor, for its intuitiveness, simplicity and clarity, for its coherence with the ancient tradition of geometry exemplified by Euclid, and for the role it gave to mathematical genius. They opposed synthesis to what they characterized as the analysts' blind application of algorithms. Mazzotti shows how members of this school were also political reactionaries, supporting the continuation of feudalism and its attendant monarchy. He makes the case that these two reactions were two sides of the same coin: both geometry and politics should respect past orders, that of Euclid or of the established Neapolitan ways of organizing agriculture and society more generally. The impulses toward both reactions were the same, and in this way, the two reactions reinforced each other.

The book is a splendid opening to an approach to the study of mathematics that not only situates it historically but also politically. Indeed, Mazzotti stresses that mathematical knowledge is essentially historical, that its objects and practices are "essentially situated in space and time, shaped by the circumstances of their creation and use" (p. 9). These circumstances are in part political and attention to them is accordingly necessary for a full accounting of mathematical knowledge.

Nonetheless, I have some reservations about the work. Firstly, the looseness of its analysis of the notion of purity occludes some points that could have been made, either more sharply or at all. As we have seen, Mazzotti characterizes purity in terms of neutrality. But sometimes neutrality is understood as universality, wherein analysis applies to any mathematical subject-matter; and sometimes it is understood as disinterestedness, as in the engineers who apply analysis to problems without concern for local conditions. These are not the same. Geometers since Descartes tried to develop analytic methods that were at the same time universal and yet attentive to the particulars of the problems being solved; Möbius and Plücker were pursuing this roughly contemporaneously with the Neapolitan episode of this book (cf. Tappenden, 2005, pp. 57-8).

Mazzotti has the resources to make this distinction clearer. In discussing the changes envisioned by the French occupiers, he observes that

Critics of reform lamented the excessive bureaucratization of the new administration, which they portrayed as too expensive and detached from the concrete reality of the country. To them, reformist tenets were French and alien, guided by abstract egalitarian norms that conflicted with both local traditions and common sense. Provincial borders had been redrawn in ways that ignored local history and customs (p. 77).

Later, he introduces the work of another member of the synthetic school, Francesco Bruno:

> But Bruno's masterpiece, according to his admirers, were his solutions to a group of plane-geometry problems: they were considered particularly elegant because he used "loci
whose description and determination is the simplest possible one” and which could be di-
rectly extracted from the data, so that no reference was made to geometrical truths exter-
nal to the problem, thus preserving its true unpolluted nature – a local nature. In this sense
Bruno’s solution is “more natural” than other possible geometrical solutions (p. 161).

These two points could have been woven together. The reactionary approach to organizing society required strict attention to local conditions. Careful attention to the particulars of the land, its rivers and its hills, was needed in order to organize its cities, roads and bridges. Since locals knew the land better than foreigners, the reaction called for opposition to the French occupiers who by contrast favored an analytic, abstract approach to engineering. We can see the same contours in the debate about mathematics, as in Bruno’s solutions that used only what was “internal” to the problem. It would have been better had Mazzotti drawn this parallelism explicitly.

Here the microhistorical approach may be responsible. Purity as a preference for the local has a long history in mathematics, going back to Aristotle and Euclid, arising in a transformed way with Descartes' application of algebra to geometry and the reaction against this application by Newton. A preference for the local in mathematics has persisted until the present day, even if, as in earlier times, this preference competes with preferences for more global, unifying approaches. Purity as a preference for the local has been in recent years studied by philosophers of mathematics (see my Arana 2024 for an overview). Mazzotti's focus on his case study, laudable in many ways, may have shielded him from considering the wider context of purity, both historically and analytically.

Relatedly, as noted earlier, this very same debate, between analytic and synthetic approaches to geometry, was happening simultaneously elsewhere. Even the French, posed in this book as bearers of the analytic revolution, had the same debate, and had been since Descartes. In the early 19th century, Poncelet and Gergonne in France, and Plücker in Germany, were trying to reckon with analytic approaches to projective geometry that could at the same time make use of the geometrical specificities of the problems being solved. The 2015 article by Lorenat is a nice discussion of this debate, and indeed Mazzotti cites it, but only in commenting briefly that a member of the Neapolitan synthetic school and student of Fergola, Francesco Paolo Tucci, made some remarks on this French debate. Mazzotti does not tell us what Tucci said, however. Placing the Neapolitan debate within this French (and German) context might have improved our understanding of the particularities of the Neapolitan position in this debate, perhaps bringing out new insights into preferences for algebraic and geometric ways of thinking.

A final remark of this sort concerns Mazzotti's framing of mathematical puri-
ty as a return to order, notably in the title of Chapter 8, the book's last. The idea is
that synthetic approaches to geometry reflect the "natural order" of geometry in
their attention to the local, a reflection that carries over to their role in sustaining
the traditional moral order. But analytic geometry is also ordered, in an even more
explicit way; for each algebraic equation has a degree, and this gives geometers a
quantitative way to order the complexity of the curves represented by such equa-
tions. This idea goes back to Descartes, as Bos has shown (cf. Bos, 2001). It is not
that the synthetic school sought order *tout court* in its preference for purity, but
rather, that it sought a particular kind of order, namely, the order perceived by
ancient geometry (and moralists).

These remarks are meant to suggest improvement to what is already a major accomplishment. Mazzotti's sustained treatment of the interlacing of mathematics and politics, which becomes evident once we recognize mathematics as historical, is fascinating and worth careful study. It is on the vanguard of a new and much-needed shift in perspective on mathematical practice.

ANDREW ARANA

Université de Lorraine and Archives Poincaré (UMR 7117 CNRS)

andrew.arana@univ-lorraine.fr

REFERENCES

ARANA Andrew, 2024. *Elements of purity*. Cambridge University Press, Cambridge.

ARANA Andrew, BURNETT Heather, 2023. Mathematical hygiene. *Synthese*, 202/110, pp. 1-28.

Bos Henk J.M., 2001. *Redefining Geometrical Exactness*, Springer, New York.

LORENAT Jemma, 2015. Figures real, imagined, and missing in Poncelet, Plücker, and Ger-
gonne, *Historia Mathematica*, 42/2, pp. 155-192.

MACINTYRE Alasdair, 1990. *Three Rival Versions of Moral Enquiry*, University of Notre Dame Press, Notre Dame.

TAPPENDEN Jamie, 2005. "Proof style and understanding in mathematics. I. Visualization, unification and axiom choice," in Paolo Mancosu, Klaus Frovin Jørgensen, and Stig An-dur Pedersen (eds.), *Visualization, explanation and reasoning styles in mathematics*, Sprin-ger, Dordrecht, pp. 147-214.

CESARE LOMBROSO, 2022. *L'amore nei pazzi e altri scritti (1851-1909)*, a cura di Alberto Cavaglion, con un saggio di Silvano Montaldo, Torino, Einaudi, 720 pp.

At the International Congress of Anthropology held in Geneva from 24 to
29 August 1896, following Cesare Lombroso's speech, the Argentinian sociologist
José Ingenieros said of him: "Es un hombre genial, pero no es intelligente" (He is
a brilliant man, but not intelligent). This seemingly paradoxical and unflattering
statement, made by someone who considered himself his friend, encapsulates
all of Lombroso's ambivalence. This ambivalence was, indeed, the hallmark of
the Positivist age, that Lombroso represented and reflected. An exceptional case
in Italian cultural history for the dissemination of his works, which made him
the most famous Italian in the world between the late 19th and early 20th cen-