Abstract
A well known conception of axiomatization has it that an axiomatized theory must be interpreted, or otherwise coordinated with reality, in order to acquire empirical content. An early version of this account is often ascribed to key figures in the logical empiricist movement, and to central figures in the early “formalist” tradition in mathematics as well. In this context, Reichenbach’s “coordinative definitions” are regarded as investing abstract propositions with empirical significance. We argue that over-emphasis on the abstract elements of this approach fails to appreciate a rich tradition of empirical axiomatization in the late nineteenth and early twentieth centuries, evident in particular in the work of Moritz Pasch, Heinrich Hertz, David Hilbert, and Reichenbach himself. We claim that such over-emphasis leads to a misunderstanding of the role of empirical facts in Reichenbach’s approach to the axiomatization of a physical theory, and of the role of Reichenbach’s coordinative definitions in particular.
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Notes
For a recent account of the fundamental methods of logical empiricism, see Lutz (2012).
The abstract or strictly formalist reading emphasizes the fact that Hilbert’s axiomatization of geometry provides “implicit definitions” of the terms of geometrical theories (“point”, “line”, “plane”), which do not specify a unique referent for these terms. The term “implicit definition” was not often used by Hilbert himself, despite its centrality to formalist readings of Hilbert (our thanks to an anonymous reviewer for this point).
Hilary Putnam (1974) argued that Reichenbach’s view does not show how relations between abstract and physical elements of theories are controlled. If the axioms and definitions of a theory are arrived at by free choice, and if they specify the class of models under which the claims of the theory are mapped to “true”, then it seems that the fundamental relations of the theory float free from the theory’s empirical content. Lionel Shapiro argues in response that “Reichenbach's much-maligned 'conventionalism' reduces to a forceful illustration of the fact that mathematical structures remain physically contentless pending specification of the real-world correlates of their abstract elements” (Shapiro, 1994, p. 296).
Unsurprisingly, this is particularly applicable to modern theoretical physics. In the case of general relativity in particular, Michael Friedman has argued that “the four dimensional, variably-curved geometry of general relativity is an entirely non-intuitive representation having no intrinsic connection whatever to ordinary human sense experience”, and that this prompts “both the logical empiricists and Einstein himself” to “discern an intimate and essential relationship between the general theory of relativity, on the one hand, and the modern ‘formal’ or ‘axiomatic’ conception of geometry associated with David Hilbert, on the other” (Friedman, 2001, p. 78).
Richardson (2021) suggests a reading of Reichenbach’s (1920) The Theory of Relativity and A Priori Knowledge according to which “a scientific method in philosophy would show that attention to the formal features of knowledge in the exact sciences of nature requires that philosophy of science move from scientific neo-Kantianism to a theory of knowledge that is closer to empiricism. This was the main argument of the 1920 book” (p. 158). Our reading of Reichenbach’s (1924) book is consistent with this account of his philosophical development at the time (without attributing our reading to Richardson). Richardson’s account of Reichenbach’s development after 1920 (and the influence of Schlick) is significant context for the 1924 book.
In his student days, “Reichenbach studied mathematics, philosophy, and physics under such teachers as Born, Cassirer, Hilbert, Planck, and Sommerfeld at the Universities of Berlin, Göttingen, and Munich” (Salmon, 1979, p. 5), and in 1918-1919, Reichenbach “was one of five intrepid attendees of Einstein’s first seminar on general relativity given at [the Humboldt-Universität in Berlin] in the tumultuous winter” (Ryckman, 2018, Sect. 4.1).
See Eder and Schiemer (2018), Sect. 2.3.
“The two main aspects of Pasch’s philosophy are a formal stance with regard to the validity of mathematical deductions and a strong commitment to an empiricist understanding of the basic concepts of mathematics” (Schlimm, 2010, p. 94). While, as Schlimm notes, these two elements may seem to be in tension with each other, the basic aim of Pasch’s methods of axiomatization is to distinguish them carefully.
Here we follow Schlimm (2010).
As Pasch notes in his early work on the principle of duality in projective geometry, formal concepts may refer to more than one thing, and axioms may be duals (equivalent, but distinct formal statements of an axiom with different content). Thus, without further elaboration, neither can be used to provide a definition of the core concepts of a theory, only the stem concepts. See (Pasch, 1914, p. 143) and discussion in (Schlimm, 2010, p. 100). See (Eder & Schiemer, 2018, Sects. 2.2 and 2.3) for the principle of duality and its importance for Pasch and Hilbert.
Note that Hertz himself did not use the expression “configuration space”. For some relevant discussion of this point, see Lützen (2005, pp. 129–131 and 154–156).
At the close of his preface, Hertz writes, “What I hope is new, and to this alone I attach value, is the arrangement and collocation of the whole [die Anordnung und Zusammenstellung des Ganzen]—the logical or philosophical aspect of the matter. According as it marks an advance in this direction or not, my work will attain or fail of its object.” (Hertz, 1894/1899, p. xxiv).
“Mature knowledge regards logical clearness as of prime importance: only logically clear images does it test as to correctness; only correct images does it compare as to appropriateness. By pressure of circumstances the process is often reversed. Images are found to be suitable for a certain purpose; are next tested as to their correctness; and only in the last place purged of implied contradictions” (Hertz, 1894/1899, p. 10).
In this sense, the tradition of empirical axiomatization provides a way to deepen our understanding of Reichenbach’s well-known distinction between the contexts of discovery and of justification in Experience and Prediction (1938). Theories develop in an organic and messy way, and axiomatization clarifies the theory, analyzes it into its components, and elucidates the relations of dependence between the elements of the theory. It is possible that Reichenbach’s contact with the tradition of empirical axiomatization informed the development of his view on discovery versus justification, over the 14 years between ATR and Experience and Prediction. For more on discovery and justification, including on the historical development of Reichenbach’s views, see Schickore and Steinle (2006), perhaps especially the essay by Gregor Schiemann.
Hilbert offered the following vivid metaphor for the role of axiomatization in the development of science: “The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development” (Hilbert, 1905, p. 102; quoted from translation in Corry (2018, p. 5).
As Hilbert says in a letter to Frege: “it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney sweep, … and then assume all my axioms relations between these things, then my propositions, e.g. Pythagoras' theorem, are also valid for these things. In other words: any theory can always be applied to infinitely many systems of basic elements.” Letter from Hilbert to Frege, 1899; citation from excerpt in Frege (1980, pp. 40–41). Notably, Moritz Pasch also uses the metaphor of “scaffolding” in discussing geometrical concepts; see the section on Pasch above and Schlimm (2010, p. 103).
As Hilbert put the matter: “The present investigation is a new attempt at formulating for geometry a simple and complete system of mutually independent axioms; it is also an attempt at deriving from them the most important geometric propositions in such a way that the significance of the different groups of axioms and the import of the consequences of the individual axioms is brought to light as clearly as possible” (Hilbert, 1899/2004, p. 436).
“Pasch’s lectures exerted a considerable direct influence on Hilbert’s thinking about geometry and axiomatics in general, as can be seen from the development of Hilbert’s lecture notes on geometry in the 1890s, which contain lengthy paraphrases of Pasch’s discussions, and also from remarks Hilbert made in his correspondence. This deep influence is not acknowledged properly in Hilbert’s seminal Grundlagen der Geometrie (1899), where Pasch is only credited in a footnote for the first ‘detailed investigations’ of the axioms of betweenness, in particular the axiom that became later known as Pasch’s axiom’” (Schlimm, 2010, p. 93).
The terminology “complete and simple” reflects the influence of Hertz’s Principles on Hilbert; see Patton (2014) for details. In a similar vein, just as Hertz had constructed a logically perspicuous “image” (Bild) of mechanics, Hilbert sought a complete image of “geometric reality”; see Sieg (2020), p. 14.
We are grateful to an anonymous reviewer for insightful suggestions that materially improved this section.
Hilbert’s First Communication was submitted to the Göttingen Academy of Science for publication in its Proceedings in November 1915 (1915/2009), and his Second Communication was submitted to the Academy a year later in December 1916 and published in 1917 (1917/2009). In 1924, Hilbert combined both papers into a single article to be printed (with some revisions) in the Mathematische Annalen—see Majer and Sauer (2009, p. 26) (and also Brading and Ryckman, 2008, p. 103).
“The central idea of Hilbert's communication is to combine both Einstein's [general relativity] and Mie's [electrodynamic theory] in a variational framework with an invariant action integral. In the paper, Hilbert showed how to generalize Lorentz-covariant electrodynamics to a generally covariant theory in such a way that… generally covariant gravitational field equations follow from the variation of the action integral” (Sauer, 2002, p. 225). And: “In seeking a derivation of the field equations of gravitation from a variational principle, Hilbert upped the ante in postulating a single generally invariant ‘world function’, a Lagrangian for both the gravitational and the matter fields, from which the fundamental equations of a pure field physics might be derived” (Brading & Ryckman, 2008, p. 110).
See Sauer (2006), pp. 227–228 and passim.
Of course, we may be skeptical of whether Hilbert succeeded. For historical purposes, that question is less important than what Hilbert thought he was doing.
Hilbert (1915/2009, pp. 45–46; trans. Sauer 2006, p. 228).
As Brading and Ryckman note, “set within the logical and epistemological context of the ‘axiomatic method’, Hilbert’s two notes may be seen to have the common goal of pinpointing, and then charting a path toward resolution of, the tension between causality and general covariance that, in the infamous ‘hole argument’, had stymied Einstein from 1913 to the autumn of 1915. Unlike Einstein’s largely informal and heuristic extraction from the clutches of the ‘hole argument’, Hilbert stated the difficulty in a mathematically precise manner as an ill-posed Cauchy problem in the theory of partial differential equations, and then indicated how it can be resolved” (2008, p. 104).
Hilbert’s axiomatic approach was “above all, a tool for retrospectively investigating the logical structure of well-established and elaborated scientific theories” (Corry, 2018, p. 5; emphasis in original). Corry (2018) has argued that it is important to distinguish between a narrower and broader conception of Hilbert’s “formalist program”. On the narrower construal, Hilbert’s program is mainly concerned with demonstrating specific results like the consistency of mathematics using finitist arguments. On the broader construal, Hilbert’s program carries with it an overarching conception of the essence of mathematics as only a system of conventional rules applied to arbitrary signs. Reichenbach’s own reading of Hilbert may be due to the logical empiricists’ focus on Hilbert’s metamathematical program; see the discussion in Majer (2002) and Sieg (2020).
For a historical discussion of Reichenbach’s relationship to Hilbert, including the “epistemological and institutional connection between Reichenbach’s circle in Berlin and Hilbert’s in Göttingen” (p. 33), see Benis Sinaceur (2018), Sect. 3.
Majer contributes a nuanced analysis of Hilbert’s research programs, and of the impact the logical empiricists’ misunderstanding of Hilbert’s aims for the axiomatic method had on their understanding of his epistemological achievements (Majer, 2002, pp. 218–222). It is beyond the scope of this paper to discuss these matters further here. Majer’s paper does not focus on Reichenbach in particular, thus, our paper could provide a useful complement to Majer’s analysis. See also Mayer 1995 for details on the logical empiricists’ reception of Hilbert.
Hilbert focused on the New Foundations and on metamathematics from 1918 to about 1922; see Majer (2002, p. 218).
This is supported by Majer and Sauer’s introduction to the work (Majer & Sauer, 2009, p. 378).
We are grateful to an anonymous reviewer for raising this point.
“In the nineteenth century, logicians viewed the consistency of a notion from a semantic perspective as requiring a model. That is the way we put matters, whereas those earlier logicians, including Frege, saw themselves as facing the task of exhibiting a system that falls under the notion” (Sieg, 2020, p. 138).
As Majer notes, this is a shame: “the logical empiricists would have done better, if they had paid somewhat more attention to Hilbert and his axiomatic approach to science” (2020, p. 213).
Our thanks to an anonymous reviewer for prompting us to elaborate on this point.
Although we do not discuss Reichenbach’s treatment of general relativity in this paper, see Ryckman (2005) Section 4.4.5 for a criticism of Reichebach’s “preposterous attempt to derive general relativity over finite regions from the infinitesimal validity of special relativity.”.
Note that the issue of a deductive vs. constructive approach to axiomatization is orthogonal to the issue of an empirical vs. formal approach.
“[T]his investigation starts with elementary facts as axioms; all are facts whose interpretation can be derived from certain experiments by means of simple theoretical considerations. Such a treatment (or separation) presents itself all the more readily, since the theory whose foundation is the subject of this book—Einstein’s theory of relativity—constitutes an innovation in physics, and the elementary facts can be chosen in such a way as not to presuppose the new theory for their interpretation in connection with the experiments on which these facts are based” (ATR, 6).
See Ryckman (2005, pp. 99 and 105) for a brief discussion of the oddities of Reichenbach’s use of the term “topological” in this context.
Note, however, that Reichenbach is careful to use the expression “Fermat’s axiom” rather than “light principle” to label Axiom III; cf. ATR §21.
It is not until Definition 8 that Reichenbach introduces the standard value ε = 1/2, a stipulation which he labels “Einstein’s definition of simultaneity” (ATR p. 44).
Recall that, although there may be good reasons to revise these definitions, they cannot themselves be evaluated with regard to their truth or falsity (they are “neither true nor false”).
Note that there is a narrower view of formalism according to which Pasch, Hilbert, and Reichenbach would all be formalists. See Corry (2018).
It is also important to note that Reichenbach’s discussion of coordination in 1920 is preliminary. For one thing, Reichenbach has not yet articulated the expression “coordinative definition.” Furthermore, the thrust of his discussion is concerned with the peculiar problem of comparing a mathematical structure with reality (in contrast with the well-defined problem of comparing two mathematical structures with one another); a problem which he addresses, early on, by fleshing out his distinctive neo-Kantian framework.
Our thanks again to an anonymous reviewer for highlighting this point.
Similarly, one might argue that the axiomatic clarification of a theory, or the conventional specification of definitions or axioms, takes place independently of empirical reasoning. The above discussion focuses on how, for Reichenbach’s axiomatization of relativity, these are deeply entwined.
The 19th and early twentieth century ‘empirical axiomatization’ we explore need not be read as ‘empirical’ in the sense employed in more contemporary theories. None of these figures require that the axioms of a theory be reduced entirely to material claims, for instance. Instead, the empirical tradition at stake in this paper begins with a mature theory and goes on to clarify which of its content is abstract or formal, and which is empirical—and which of the results of the theory depend on which. Throughout this process of analysis and clarification, it is necessary to appeal to empirical content—which can include concrete ‘intuition’, observation, and experiment.
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Acknowledgements
We are grateful for detailed comments on earlier drafts of this paper by Alison Fernandes and James Read, as well as to generous and helpful comments by anonymous reviewers for this journal. Our thanks also to Erik Curiel and Flavia Padovani for their discerning editorship of this Special Issue and for their editorial guidance.
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Eisenthal, J., Patton, L. Reichenbach’s empirical axiomatization of relativity. Synthese 200, 464 (2022). https://doi.org/10.1007/s11229-022-03936-z
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DOI: https://doi.org/10.1007/s11229-022-03936-z