Realism, Functions, and the a priori: Ernst Cassirer's Philosophy of Science



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"Realism, Functions, and the A Priori: Ernst Cassirer's Philosophy of Science"

Jeremy Heis, University of California–Irvine

Version 3.0

31 May 2012


ABSTRACT: This paper presents the main ideas in Cassirer's general philosophy of science, focusing on the two aspects of his thought that – in addition to being the most central ideas in his philosophy of science -- have received the most attention from contemporary philosophers of science: his theory of the a priori aspects of physical theory, and his relation to scientific realism.

Ernst Cassirer, the Neo-Kantian trained philosopher whose wide-ranging work spanned the first four decades of the twentieth century, was one of the most prominent and respected philosophers of science of his time. Easily the most subtle and mathematically well-informed of the Neo-Kantians, he was among the vanguard of early twentieth century philosophers seeking to understand the philosophical significance of the revolutionary advances made in logic, mathematics, and physics. Not only did Cassirer write some of the earliest philosophical works on general relativity and quantum mechanics,1 but he was one of the first German academic philosophers to give serious attention to Russell's logicism and the new logic,2 Dedekind's foundations of arithmetic, and to Hilbert's axiomatic foundation of geometry.3 Cassirer also wrote extensively on some of the perennial issues in general philosophy of science: realism, confirmation, theory change, the nature of experimentation, the a priori elements in scientific theories, and the application of mathematics in physical science. Cassirer's commitment to a philosophy of science that engaged with cutting edge science ran deep and was widely known. For example, as a letter from Hans Reichenbach makes clear, Cassirer was the only philosopher to sign onto a petition, composed by Reichenbach in 1931 and co-signed by Hilbert and Einstein, petitioning the German government to create a chair in the philosophy of science.4

It is not surprising, then, that Cassirer's work has been studied extensively by historians of philosophy. But several prominent philosophers of science have also recently turned their attention to Cassirer, finding in his writings philosophical positions that only now many years later are receiving sustained philosophical interest. For example, many defenders of “structural realism” within the philosophy of science have explicitly pointed to Cassirer’s writings as an historical anticipation of their own theories,5 and Michael Friedman has identified Cassirer as an inspiration for the philosophy of science defended in his book Dynamics of Reason.6

Despite this attention, there is still disagreement among philosophers over the interpretation of many of the main ideas of Cassirer's philosophy of science. For example, one of the most striking and suggestive features of his philosophy is its conception of the a priori. Many interpreters have claimed that Cassirer's theory of the a priori is an anticipation (and, indeed, the historical source) of the theory of the relativized, but constitutive a priori later articulated by Reichenbach (who was, after all, Cassirer's student in Berlin). According to Michael Friedman, however, the a priori elements that Cassirer claims to find in science are absolute, but not relative, and are merely regulative, not constitutive (Friedman, A Parting of the Ways, 115ff). Alan Richardson, on the other hand, argues that Cassirer, in trying (unsuccessfully) to integrate a theory of the constitutive, relative a priori with a theory of the regulative, absolute a priori, succeeds only in presenting a "plurality of inconsistent accounts" (Richardson, Carnap’s Construction of the World, 136). Last, Thomas Ryckman argues that for Cassirer (in his book Einstein's Theory of Relativity) the principle of general covariance is both constitutive and regulative (Reign of Relativity, 46) – thus agreeing with Richardson (against Friedman) that there are constitutive a priori principles in Cassirer's theory, while disagreeing with Richardson that this amounts to any kind of tension in Cassirer's thinking. Similar issues arise in discussions of Cassirer's relation to realism. As French, Ladyman, and Massimi all recognize,7 there is a clear (though hard to articulate) tension between a reading of Cassirer's philosophy that brings him close to contemporary structural realism, and Cassirer's own rejection of realism in favor of idealism.

The goal of this paper is to present the main ideas in Cassirer's general philosophy of science. In particular, I will present the contours of the two aspects of his thought that – in addition to being the most central ideas in his philosophy of science -- have received the most attention from contemporary philosophers of science: his theory of the a priori, and his relation to scientific realism. I will argue (against Friedman) that Cassirer's theory assigns both constitutive and regulative, relative and absolute, roles for a priori representations. These a priori representations help explain the possibility of scientific objectivity and thus also objective reference. This theory of objectivity, and the need to secure it with a priori elements, flows out of the distinction between substance and function that is the main theme of Cassirer's 1910 book Substance and Function. In section I of this paper I describe that distinction. In section II, I describe Cassirer's two part theory of the a priori and argue – contrary to Richardson – that Cassirer's second, absolute, part of the theory of the a priori is not inconsistent with the first, relativized, part, but necessitated by it.

The role of the theory of the a priori in Cassirer's philosophy of science is in explaining physical objectivity. In particular, objectivity is maintained in science even as physical theories change because the structure of science remains the same even as the fundamental ontologies of theories are replaced by successor theories. This naturally raises the philosophical question whether Cassirer was a "realist" at least about the structural features of physical theories, as are contemporary structural realists. In the last section of this paper, I critically evaluate this claim, distinguishing the senses in which Cassirer's philosophy of science is (and is not) a form of "realism."

I. "Substanzbegriff" and "Funktionsbegriff"
The animating idea in Cassirer's philosophy of science – indeed, in his theoretical philosophy in general – is the contrast between substance-concept [Substanzbegriff] and function-concept [Funktionsbegriff], a contrast that provides the title for his first systematic book-length work in the philosophy of science.8 Cassirer means this contrast to cover a number of different interrelated epistemic, logical, and metaphysical contrasts. Of these many uses, the most fundamental use that Cassirer makes of "Substanzbegriff" and "Funktionsbegriff" is epistemological and Kantian: it contrasts philosophical views that overlook the epistemic preconditions of various kinds of knowledge, with those that recognize the "functions" [or "preconditions"] that make certain kinds of knowledge possible. To oppose the point of view of "Substanzbegriff," then, is to oppose various forms of epistemological atomism: the view that certain kinds of knowledge (be they scientific concepts, experiences, or measurements) could be acquired all by themselves, without any other epistemic conditions.

Cassirer's paradigm example of "Substanzbegriff," is an atomistic theory of concept acquisition (thus the phrase "substance-concept" to describe the approach he rejects). But the fundamental kind of atomism that he wants to oppose in the philosophy of physical science concerns measurement.9 After analyzing a series of cases of measurement in physical science, Cssirer argues that these most basic results of scientific experimentation presuppose not only concepts and laws of pure mathematics, but also laws of nature and natural scientific concepts. Here are some of the cases Cassirer discusses: assigning a real number to a temperature by measuring a volume of mercury presupposes laws of geometry and the law relating temperature to the expansion of volume of mercury (142-3); when Regnault measured the volume of a gas in his tests of Boyle's law, he used an instrument the design of which just presupposes the "abstract principles of general mechanics and celestial mechanics" (143; cf. Duhem 1906, 145-7); the measurement of time requires the identification of a unit, the choice of which presupposes the law of conservation of energy, or the principle of inertia (145; cf. Poincaré 1905, 210-22); the measurement of the curvature of space presupposes the choice of a "rigid body" (107; cf. Poincaré 1902); the determination of the position of a body requires constructing a Langean "inertial system" (182); Ampère's measurement of the intensity of an electric current required a galvanometer, whose operations presuppose various physical laws (SF, 280).

The failure of epistemological atomism about measurement, on Cassirer's view, generalizes to all of our scientific knowledge. Since even these most elementary experimental results presuppose a system of mathematical and theoretical concepts and even certain theoretical laws, Cassirer concludes that no scientific concept or principle could be acquired all on its own. Moreover, as Cassirer learned from Duhem, no single empirical statement of natural science can be confirmed atomistically.10

Cassirer believes that these non-atomistic conceptions of measurement and empirical confirmation require a distinctive, "functional" theory of objectivity. Since it is impossible to know any fact about a physical object outside of a system of concepts and principles, the objectivity of a particular scientific claim cannot be determined except by answering the prior question of the objectivity of a total theory. This motivates for Cassirer the view that objectivity is primarily a property of a total theory.11 Cassirer explicates the notion of objectivity of a total theory using two further notions: unity and permanence.12 "Unity" for Cassirer includes not only logical coherence (both among the various sentences in a theory and with the measurements that provide its empirical support), but also a certain systematic form. This form is provided by a theory's laws, which logically structure the total theory.

The significance of "permanence" or "constancy" for Cassirer's conception of objectivity is difficult to overstate. As Cassirer puts it, "'objective' ['Gegenständlich'], in the critical sense of the word, means that which is 'constant' ['beständig'] in our cognition."13 This permanence operates on multiple levels. In the simplest case, the properties of an individual object (say, its spatial location with respect to other objects in its environment) are objective, while the various appearances of that object to different observers are subjective, since the appearance of an unchanging object can alter as an observer moves about with respect to it. The permanence of the observed object makes itself manifest in the constant law that would predict its appearance as a function of the position of the observer. Here the "law" is objective, while the appearances are not.14 Particular facts about individual objects can then be united into a total theory by means of higher level physical laws that describe the behavior of physical objects over time. Though these higher laws are permanent in their own sphere (the laws giving, say, the forces acting on a moving body are permanent even as the body constantly changes its position and velocity), still no physical theory will be the final word. The objectivity of the revision or replacement of the highest laws in a physical theory is a particularly pressing issue, since (following Duhem) Cassirer allows that there could be two distinct theories equally compatible with the empirical evidence. For this reason, Cassirer argues that there are supreme "laws" or norms that guide the formation and selection of new physical theories. Science can remain objective, then, even as theories come and go, since there are permanent laws shaping this process of theory formation and selection.15

Cassirer builds on this "functional" theory of objectivity to give a functional theory of knowledge and truth: A representation is knowledge if it plays the sort of role within a system of representations that would make objectivity possible.16 The concept 17, finally, is understood in terms of and , as that which is represented by fully objective knowledge.18 Cassirer emphasizes that the conceptual ordering among the core concepts , , , and on the functional theory is opposed to the traditional ordering in the "substance" theory of knowledge. According to the "substance" or "copy" theory of knowledge, the primitive concept is the metaphysical concept of an .19 is explained in terms of the concept , where a representation is true if it faithfully mirrors the properties of objects. Knowledge is then a certain species of true representation – a "copy" of objects, as it were. Last, objective knowledge is a kind of knowledge whose special status is explained in terms of the peculiarities of its object: objective knowledge is about "external" objects, not the inner states of a subject.


II. Theory Change and the A Priori
The fundamental concept of the functional theory of knowledge is objectivity, and objectivity is understood in terms of the "unity" and "permanence" of physical theory. On Cassirer's view, there is a system of a priori concepts and principles that are conditions of the possibility of this unity and permanece. In this section, I will outline Cassirer’s theory of the a priori and show why Cassirer considered it to be a consequence of the polemics against epistemological atomism (and the "substance theory of knowledge") I've just described.

I noted above that the interpretation of Cassirer's conception of the a priori has been a matter of dispute. Some have argued that it anticipates liberalized or relativized versions of the theory of a priori, such as the theory defended by Reichenbach in his The Theory of Relativity and A Priori Knowledge and revived more recently by Michael Friedman.20 Others, most notably Friedman himself, have denied that Cassirer's theory includes relative, but constitutive a priori principles.

I believe that there is ample textual evidence that Cassirer does in fact maintain that there are relativized, constitutive a priori principles:

That we in [science] find only a relative stopping point, that we therefore have to treat the categories, under which we consider the historical process itself, themselves as variable and capable of change, is obviously correct: but this kind of relativity does not indicate the limits, but rather the particular life of cognition.21


And at ETR 415, he gives examples of these a priori but nevertheless changeable principles.

That a step is thereby taken beyond Kant is incontestable; for he shaped his “Analogies of Experience” essentially on the three fundamental Newtonian laws; the law of inertia, the law of the proportionality of force and acceleration, and the law of the equality of action and reaction. But in this very advance the doctrine that it is the “rule of the understanding”, that forms the pattern of all our temporal and spatial determinations, is verified anew. In the special theory of relativity, the principle of the constancy of the velocity of light serves as such a rule; in the general theory of relativity this principle is replaced by the more inclusive doctrine that all Gaussian coordinate systems are of equal value for the formulation of the universal natural laws. It is obvious that we are not concerned here with the expression of an empirically observed fact, but with a principle which the understanding uses hypothetically as a norm of investigation in the interpretation of experience. (ETR, 415)


Though these principles are a priori (they are not "expressions of an empirically observed fact"), Cassirer nevertheless explicitly denies that they are apodictic, certain, or self-evident, and he denies that we have any conclusive reason to think that even the constitutive principles of the general theory of relativity22 will be constitutive in all of our future physical theories.23 These principles (and the concepts they contain) are therefore relative but constitutive a priori.

However, this relativization of Kantian categories seems prima facie to be radically at odds with Cassirer's theory of objectivity. As I emphasized above, Cassirer strongly associates objectivity with permanence: “We finally call objective those elements of experience, which persist through all change in the here and now, and on which rests the unchangeable character of experience" (SF 273). For this reason, as I noted, Cassirer thinks that the objectivity of experience is grounded in physical laws, since – though physical objects may change their properties over time – the laws describing these changes are permanent. Granted: these relativized constitutive a priori principles play an essential role in grounding the objectivity of experience, since lower level laws that describe the behavior of objects (like the laws of planetary motion) are made possible by constitutive principles (like Newton's principle of inertia).24 However, if the very highest laws of our physical theories are not permanent but only relative, then the very possibility of physical objectivity seems threatened.

For this reason, it is of the utmost importance to Cassirer to identify elements in physical theories that do remain the same even as the fundamental concepts and principles (the relative, constitutive a priori elements) evolve. In a series of passages in SF (268-70; 321-2), Cassirer identifies seven such permanent elements:


  1. Mathematical concepts and principles (though which mathematical concepts and principles get employed in a given theory may change).

  2. Some questions posed by the older theory (which get answered in the new theory). For example, the question Why do planets travel in elliptical orbits? could be posed in Kepler's day and only later answered by Newton.

  3. Some empirical facts (though they are interpreted in a new way in the new theory). For example, Tycho's observations of the position of Mars can be carried over from the Ptolemaic to the Newtonian system.

  4. “Ultimate invariant” concepts, like ,

  5. The mathematical form of the old theory (which is maintained at least as a limiting case in the new theory).25 For example, Newton's laws become approximately true in the small in the theory of relativity.

  6. Some principles of theory selection.

  7. The principle of the “unity of nature.”

Numbers 2, 3, and 5 are empirical; 1, 4, 6, and 7 are then a priori, though in an absolute, not relative sense.

Cassirer describes these a priori concepts and principles as elements that remain "invariant" throughout the entire history of science.

The goal of critical analysis would be reached, if we succeeded in isolating in this way the ultimate common element of all possible forms of scientific experience; i.e., if we succeeded in conceptually defining these moments, which persist in the advance from theory to theory because they are the conditions of any theory. At no given stage of knowledge can this goal be perfectly achieved; nevertheless, it remains as a demand, and prescribes a fixed direction to the continuous unfolding and evolution of the systems of experience.

From this point of view, the strictly limited meaning of the “a priori” is clearly evident. Only those ultimate logical invariants can be called a priori, which lie at the basis of any determination of a connection according to natural law. A cognition is called a priori not in any sense as if it were prior to experience, but because and in so far as it is contained as a necessary premise in every valid judgment concerning facts. (SF 269)

This second, absolute, sense of “priori” is then distinct from – and supplementary to – the relativized constitutive a priori elements we mentioned earlier.



As Cassirer makes clear throughout his writings, the a priori elements in this second sense are a varied lot, including: the concepts and propositions of mathematics26; and

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