Pierre Duhem’s philosophical rejection of special relativity in his 1915 pamphlet in La Science allemande [1] has been dismissed as the rantings of an aging physicist motivated by anti-German war propaganda, if not by outright anti-Semitism. While these factors undoubtedly contributed, they do not entirely account for his position.

Already in one of his early articles, L’école anglaise et les théories physiques [2], Duhem examines the national characteristics of various scientific productions. In particular, Duhem addresses the methodological differences between two schools of thought in physics, which he saw as reflective of the distinct “spirits” of the French and the English. Duhem argues that the French and English approached physical theories through fundamentally different perspectives. The French tradition, he claims, preferred a logical and deductive approach, constructing coherent, unified theories, while the English approach relied more on illustrative models, where calculations and algebraic manipulations were primary tools for navigating complex physical phenomena without necessarily achieving logical unity. This article, suitably expanded, would later form the fourth chapter of Duhem’s 1906 key epistemological work, La théorie physique [3]. Duhem famously distinguished between the deep mind of continental French and also German physics, and the ample mind of British physicists.

The deep mind favors abstraction and systematic reasoning, seeking theories based on clear postulates and rigorous logic. Deep minds emphasize the importance of developing theories from a foundation of logical postulates and rigorous reasoning. For them, calculations are not merely procedural but serve as a shorthand for logical deduction, carefully linking symbols to real physical properties. This approach strives for unity and clarity in scientific theories, forming a coherent structure of knowledge. Thinkers like Laplace, Poisson, and Ampère embody the deep mind, prioritizing coherence, clarity, and a well-ordered theoretical structure over immediate practical results.

The ample mind excels in handling intricate systems of concrete facts. Ample minds favor calculation as a primary method. For this reason, they favor fields like algebra, where symbol manipulation follows fixed rules without requiring deep engagement with underlying principles. This approach is pragmatic and thrives on symbolic algebras, such as Hamilton’s quaternions, which can condense lengthy calculations into complex operations. Figures like Lord Kelvin and James Clerk Maxwell exemplify the ample mind, as they preferred concrete representations, even if these models lacked complete logical unity.

This distinction is reflected by two kinds of modeling between continental and British thought. The continental deep mind of French but also of German physicists aims to create explanatory models that are metaphysically unified, offering a logically ordered and holistic representation of reality. Such models are constructed with clear postulates and reasoning, aiming to represent reality in a coherent way. The ample mind favors illustrative models, often employing mechanical representations. However, here the complicated systems are manipulated just as they would manipulate symbols in calculations, without rigorous logical unity. These models are pragmatic, addressing specific phenomena without necessarily aiming for comprehensive coherence.

In 1915, shortly before his death, Duhem revisited the topic of the national characteristics of different scientific productions amidst the wartime atmosphere, focusing on German science through a series of markedly nationalist articles later partially collected in La Science allemande. Here, the terms of the debate seem reversed: German scientists (previously often grouped with the French) now became the object of condemnation. In particular, Duhem seems to see the algebraization of science as a central example illustrating his broader argument about the distinct character of German science, particularly compared to French science. Duhem seems rather to have redirected his criticism of the algebraic style in physics, initially directed at British physicists, toward the Germans. However, the emphasis is different. While the British display an inclination towards algebraic manipulation at the expense of logical rigor in deduction, Duhem contends that German science aims to transform every science into a form of algebra in the name of rigorous logical deduction. If British algebraization conflicts with logic to satisfy intuition, the German algebraization of science conflicts with common sense for the sake of logical precision.

Riemann and the Algebraization of Geometry

The history of geometry serves Duhem as a blueprint for describing the process of “algebraization” of a science. Euclid‘s synthetic geometry was a deductive system based on axioms that directly addressed fundamental geometric concepts such as points and lines, and proceeded by constructing figures with those elements. Descartes’s analytic geometry is cited by Duhem as a step towards this algebraic reduction, as it translates the study of geometric figures into algebraic equations. Euclidean synthetic geometry studies figures in their own right, without recourse to formulas, whereas Descartes’s analytic geometry consistently makes use of such formulas as can be written down once an appropriate system of coordinates is adopted. However, while Descartes employs analytic methods that use coordinates, these coordinates are introduced only after the geometry is established by its own methods. Descartes’s approach, though employing algebraic tools, aimed to ensure that deductions remained consistent with an intuitive understanding of space.

Duhem argues that German science takes this further by completely eliminating any reliance on geometric intuition. This algebraic geometry, exemplified by Riemann‘s work, may achieve mathematical rigor but results in conclusions that clash with common-sense understanding of space. Riemann sought to develop a generalized theory of space beyond the Euclidean framework, and Duhem argues that he accomplished this by essentially reducing geometry to algebra. In Riemannian geometry, coordinates are introduced from the beginning, and a point is defined purely as an n-tuple of numbers; geometry’s objective then becomes to determine the geometric properties that are independent of this arbitrary choice of coordinates. Riemann defines fundamental geometric concepts—points, lines, distances—purely in terms of algebraic expressions, stripping them of their intuitive, spatial meanings derived from common sense experience. Indeed, Riemann has full freedom to define the hypothetical notion of distance between two points in an arbitrary way, recovering synthetic geometry only in special cases, such as those of constant curvature. The work of Riemann was further developed in a fully non-geometrical fashion by Christoffel, Ricci, and Levi-Civita, who created the so-called absolute differential calculus, in which geometry becomes merely one possible application. This framework consists of arrays of numbers and transformation rules to convert them from one coordinate system to another.

Common Sense vs. Good sense

Duhem could conclude that the Germans, though often highly skilled in linking together long and rigorous chains of reasoning, might frequently lack good sense or common sense. The identification of these two notions in La Science allemande seems rather puzzling. Indeed, in La théorie physique, Duhem distinguished them quite clearly. Good sense belongs specifically to scientists; it is the wisdom of the refined scientist, educated in the most modern theories, formed through long and arduous years of study. The laws of common sense are judgments that concern the general, extremely complex ideas we form about our daily observations and are independent of theory. Indeed, in La théorie physique, Duhem argues that science develops as an abstract system in contradiction to common-sense notions. On the contrary, in La Science allemande, Duhem attacks German science precisely because it disregards common sense. Since the texts are written 10 years apart, one might be tempted to simply declare Duhem inconsistent or at least sloppy. However, it is good hermeneutical praxis to rely on the presupposition of perfection, at least insofar as it remains feasible, and to assume coherence in the text or source being analyzed. With some goodwill, one can bring the two positions into agreement.

In the initial phase, where theoretical components are not at play, scientific observation is in contact with common-sense observation. However, Duhem concedes that, throughout history, scientific knowledge advances by moving away from common sense. When science is sufficiently advanced, the empirical basis in a given historical era has a solidity that depends on the theories developed in previous centuries. However, history itself represents the solid thread that maintains the connections between the most abstract theories and the world of common sense. In moving from one theory to the next, the scientist is in principle free to choose new axioms. However, their good sense should suggest to them to choose axioms such that continuity with previous theories is maintained, linking back to common-sense notions. Thus, in Duhem’s view, historical continuity guarantees that even the most refined theories have a strong anchor that reaches back to common-sense notions. German scientists lack good sense and break this historical continuity, thereby losing the connection between abstract mathematical theories and common sense. In La Science allemande, Duhem, possibly for polemical intent, simply conflates the notions of common sense and good sense, eliminating the historical separation between them: what was previously called “common sense” is now termed “esprit de finesseand the expression “common sense” is used interchangeably with “good sense.”

The history of geometry might be presented in conformity with this distinction. The fundamental concepts and axioms of geometry are of course not immediately facts of intuition, but are appropriately selected idealizations of these facts. The precise notions of, say, a point or line, for example, do not exist in our immediate sensory intuition but are only fictitious limits. The history of the synthetic approach to geometry, as seen in the work of the French school of Carnot, Poncelet, Chasles, etc., is a history of progressive abstraction from common sense, from metric to projective geometry. However, projective geometry remains a science of figures and thereby maintains the historical continuity with common-sense notions. In contrast, in the analytic approach, geometry becomes a science of numbers. Rather than refining the common-sense notion of a point, they state directly that a set of three coordinates is a point, without thinking of any particular object, and agree arbitrarily upon certain statements to hold for these points. A straight line is defined as the set of all coordinate values satisfying a linear equation, and so on. In this approach, the historical link between geometry and common sense is broken. Riemann’s ideas were indeed developed out of a purely algebraic research tradition by Christoffel, Ricci, and Levi-Civita¹.

Interestingly, Duhem presents Klein as somewhat of an exception within the German scientific tradition, stating that he “has come to assert the place of the intuitions proper to the intuitive mind in the domain of algebraic method” (86). Klein’s computational methods became the driving force behind the analytic treatment of geometric figures—that is, the shift from synthetic projection to the analytic study of coordinate transformations, furthering the development of geometry, and in particular Klein’s group-theoretical classification. However, he had the good sense to introduce such tools only after, say, synthetic projective geometry had been developed independently through synthetic methods. These methods, in turn, operate with figures, which are based on idealizations of common-sense notions. In contrast, Riemann lacked this good sense and “has tried to give to every science a form which, as much as possible, resembles that of algebra,” operating exclusively with the concept of numbers and thereby severing any link to notions of common sense, such as the intuitive notions of points, lines, etc.

Einstein and the Algebraization of Physics

Duhem acknowledged that synthetic geometry makes limited progress without the use of an appropriate language of formulas. However, he argued that analytic geometry, which completely abandons geometric representation, can scarcely be regarded as geometry. The German algebraic approach is not restricted to mathematics but also extends to physics. Germans “would like geometry, mechanics, and physics to be no more than chapters in algebra” (14). Duhem illustrated this with several examples of the “pure algebraism of German theories,” ranging from Kirchhoff to Hertz.

Duhem critiques Kirchhoff‘s presentation of physical principles in his lectures and writings. He argues that Kirchhoff presents hypotheses ex abrupto, devoid of any historical context or explanation of the intuitive processes that led to their development. This approach, according to Duhem, reduces physics to a series of algebraic deductions, neglecting the crucial role of the “intuitive mind” in shaping scientific understanding. Duhem cites a telling phrase that Kirchhoff frequently used when introducing new principles: “wir können und wollen setzen…” (we can and we will posit…). This phrase, according to Duhem, reflects the authoritarian nature of Kirchhoff’s approach, where postulates are presented as decrees, with no regard for their intuitive justification or connection to experience.

Similarly, Duhem denounces the absolute algebraism which inspires Heinrich Hertz when he claims to construct mechanics. However, his approach to electrodynamics is not different. Hertz famously declared that “Maxwell’s theory is the very equations of Maxwell.” He viewed these equations as self-evident axioms, the starting point for his investigations in electrodynamics. This approach reflects a deep trust in the power of algebra to represent and manipulate physical concepts. Hertz, like many German scientists of his time, believed that physical theories could be reduced to sets of algebraic equations, from which all other consequences could be rigorously derived. Duhem argues that Hertz’s approach, with its emphasis on algebraic formalism, became a dominant force in German science, spreading to various branches of physics and even influencing other fields like chemistry. The success of Hertz’s work on electromagnetism solidified the belief that physics could be effectively reduced to a system of algebraic equations.

However, it is the principle of relativity, as formulated by German physicists like Einstein, Abraham, Minkowski, and Laue, where the German emphasis on the algebraic mind results in a theory that defines time in a way that clashes with common sense perceptions of space and time. Indeed, here geometry itself is fully arithmetic. Relativity, according to Duhem, is a prime illustration of this tendency. Duhem criticizes the principle of relativity as a creation of the mathematical mind that goes against the intuitive understanding of space and time:

The two notions of space and time appear to all men to be independent of one another. The new physics connects them to each other by an indissoluble bond. The postulate which secures this connection is, truly, an algebraic definition of time. The principle of relativity is so plainly a creation of the mathematical mind that one does not know how to articulate it correctly in ordinary language and without recourse to algebraic formulas. It is true that Minkowski is usually regarded as the one who introduced geometrical methods into relativity; however, Minkowski ultimately treated space-time as a number manifold, as the set of all possible quadruples of coordinate values.

Duhem can then conclude that relativity, as a product of the unchecked dominance of the mathematical mind, leads to a purely algebraic theory that, while mathematically rigorous, is detached from common sense and potentially leads to physically unrealistic conclusions. Duhem thus considers both Riemann’s approach to geometry and the theory of relativity as products of an excessive “algebraism” prevalent in German science: “In such a fashion was the geometry of Riemann made. So, too, was the physics of relativity formed. In such fashion does German science progress, proud of its algebraic rigidity, looking with scorn upon the good sense of which all men have received a share.” (108). It is hard to overlook the tragic irony of expressing this criticism in 1915, the year in which Einstein published his gravitational field equations. General relativity was indeed, as Einstein put it, the triumph of the work of Riemann, Christoffel, Ricci, and Levi-Civita. This tradition is indeed purely algebraic, with minimal geometric intuition.

Einstein seemed indeed to prefer the analytic approach to geometry with respect to the synthetic one, as he confesses in a 1914 letter to Eduard Study. However, Einstein also insisted on the importance of recognizing that the historical origin of geometry in the behavior of rigid bodies, as a justification for the interpretation of relativity predictions in terms of the behavior of rods and clocks. Yet, Einstein was also aware that this stance could only provide a provisional compromise. Rods and clocks are physical systems like any others and should be ultimately described by theory itself. If this task were to be accomplished, the link between the mathematical apparatus of general relativity and geometry is indeed only historical. Far from achieving a geometry from physics, the relativistic program would have indeed expelled geometry from physics, achieving the full algebraization of physics that Duhem feared.

@book{Duhem1915,
  author       = {Pierre Duhem},
  title        = {La science allemande},
  publisher    = {Hermann},
  location     = {Paris},
  date         = {1915},
  hyphenation  = {french},
}

@article{Duhem1893c,
  author       = {Duhem, Pierre},
  title        = {L'école anglaise et les théories physiques},
  subtitle     = {À propos d'un livre récent de W. Thomson},
  journaltitle = {Revue des Questions Scientifiques},
  volume       = {17},
  number       = {34},
  date         = {1893-10},
  pages        = {345-378},
  hyphenation  = {french},
}

@book{Duhem1906,
  author       = {Duhem, Pierre},
  title        = {La th{\'e}orie physique, son objet et sa structure},
  date         = 1906,
  location     = {Paris},
  publisher    = {Chevalier et Rivi{\`e}re},
  hyphenation  = {french}
}