1 There are useful collections of in�uential papers on scienti�c realism and anti-realism in J.Leplin (ed.) Scienti�c Realism (Berkeley, California University Press, 1984) and in J. Worrall(ed.) The Ontology of Science (Aldershot, Dartmouth, 1994).

2 Duhem considered himself to be, primarily, a physicist. He made signi�cant contributionsto thermodynamics, particularly its abstract axiomatic foundations: see P. Duhem, Traitéd’énergétique ou de thermodynamique générale, 2 vols (Paris, Gauthier–Villars, 1911). Formore information on Duhem’s physics, see S. Jaki, Uneasy Genius: The Life and Work ofPierre Duhem (The Hague, Martinus Nijhoff, 1984).

3 But see A. Fine, ‘Einstein’s realism’, in J. T. Cushing, C. F. Delaney, and G. M. Gutting (eds)Science and Reality: Recent Work in the Philosophy of Science (Notre Dame, University ofNotre Dame Press, 1984), pp. 106–33; reprinted in A. Fine, The Shaky Game: Einstein,Realism, and Quantum Theory, second edition (Chicago, Chicago University Press, 1996);D. Howard, ‘Realism and conventionalism in Einstein’s philosophy of science: the Ein-stein–Schlick correspondence’, Philosophia Naturalis 21 (1984): pp. 616–29; D. Howard,‘Was Einstein really a realist?’, Perspectives on Science 1 (1993): pp. 204–51.

CASSIRER, SCHLICK AND ‘STRUCTURAL’REALISM: THE PHILOSOPHY OF THE EXACTSCIENCES IN THE BACKGROUND TO EARLY

LOGICAL EMPIRICISM

Barry Gower

1

One of the conspicuous features of current philosophy of science is the bewil-dering variety of scienti�c realisms and anti-realisms confronting us.1 This isnot so very surprising if we bear in mind that the seeds of the many relevantissues were planted at the beginning of the twentieth century, at a time ofunprecedented innovatory research in the mathematical and physical sci-ences, by scientists such as Mach, Planck, Poincaré and Duhem,2 and thatthere has been ample opportunity for the yield of those seeds to be gatheredand sifted. We have, though, done little which would help us to understandhow those seeds germinated, grew and prospered.3 In this paper I wish toexamine just one of those seeds. It has lately assumed an importance becauseof its relevance to ways in which we might restrict the ambitious scope ofscienti�c realism without curtailing that scope in an anti-realist manner.There is, I shall argue, a broader context than is usually recognized for ver-sions of restricted realism and by drawing attention to it I intend to try tobring into focus some neglected discussions which played a part in the origins

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British Journal for the History of Philosophy ISSN 0960-8788 © 2000 Taylor & Francis Ltd

British Journal for the History of Philosophy 8(1) 2000: 71–106

of logical empiricism and which are capable, I believe, of illuminating thoseorigins as well as some of our present preoccupations. More particularly, bydrawing attention to the emergence of a version of realism we now call ‘struc-tural’ realism, I shall try to support the claim that Schlick’s realism andempiricism contain signi�cant Kantian elements.4 But before turning to thehistorical issues, let me set out some of the chief features of the realism thatforms the background to this exploration of those issues.

The versions of restricted realism that I shall be concerned with count asrealisms because they claim that scienti�c theories are more than instru-mental calculating devices or co-ordinating systems, and that they aim attruth rather than mere empirical adequacy.5 Their defence needs, therefore,to identify the extra element which would contribute to the achievement ofthat aim, as well as to �nd ways of resisting the reasons which have ledothers to anti-realist positions. But at least part of their strength comes fromthe recognition by realisms’ advocates of the vulnerability of the supportcommonly offered for realism. In particular, the claim that theories in a‘mature’ science must be true, or at least approximately true, for otherwisetheir success would be miraculous, overlooks the fact that in the history ofscience successful theories have regularly turned out to be false.6 That

72 BARRY GOWER

4 The point is not, of course, new; Schlick’s Kantianism is clearly displayed in J. Alberto Coffa,The Semantic Tradition From Kant to Carnap: To the Vienna Station, edited by Linda Wes-sells (Cambridge, Cambridge University Press, 1991), ch. 9. I have discussed some aspectsof Schlick’s Kantianism in my ‘Realism and empiricism in Schlick’s philosophy’, in D. Belland W. Vossenkuhl (eds), Science and Subjectivity: The Vienna Circle and Twentieth CenturyPhilosophy, (Berlin, Akademie Verlag, 1992), pp. 202–24.

5 Scienti�c instrumentalism is the view that scienti�c theories are just calculating devicesmaking no truth claims about unobservable entities, properties and processes; their purposeis only that of enabling us to co-ordinate, classify and predict observable and experimentaldata as conveniently and correctly as possible. A more moderate kind of scienti�c anti-realism claims that although theories make claims about unobservables which aim at truth,we can never know or be justi�ed in believing that any such claim is true; we can, though,be justi�ed in believing that a theory’s claims about observable things, properties and pro-cesses are correct, in which case the theory is said to ‘save the phenomena’ or be ‘empiri-cally adequate’. For a vigorous defence of the claim that empirical adequacy is suf�cient, seeB. van Fraassen, The Scienti�c Image, (Oxford, Oxford University Press, 1980).

6 A modern version of the argument from the success of mature scienti�c theories to theirtruth is presented in R. Boyd, ‘The current status of scienti�c realism’, Erkenntnis 19 (1983):45–90. Poincaré gave a clear statement of the dif�culty for scienti�c realism presented bythe past failures of theories at the beginning chapter 10 of his Science and Hypothesis, trans-lated by W. J. Greenstreet (New York, Dover Publications, 1952), originally published as Lascience et l’hypothèse. (Paris, Flammarion, 1902):

1 The ephemeral nature of scienti�c theories takes by surprise the man of the world.Their brief period of prosperity ended, he sees them abandoned one after another; hesees ruins piled upon ruins; he predicts that the theories in fashion to-day will in a shorttime succumb in their turn, and he concludes that they are absolutely in vain. This iswhat he calls the bankruptcy of science.

(p. 160)1 For an in�uential modern statement of this argument, commonly known as the ‘pessimistic

meta-induction’, see L. Laudan, ‘A confutation of convergent realism’, Philosophy of Science48 (1981): pp. 19–48.

history is, indeed, punctuated by episodes in which earlier beliefs, includingthe most fundamental of beliefs about the kind of world we live in, are setaside in favour of new beliefs, sometimes of a revolutionary character. Typi-cally, the failure of past scienti�c theories derives from the fact that theircentral explanatory concepts are empty: there are no crystalline spheres;there is no such thing as phlogiston, or caloric; there is no aether; there areno electric or magnetic �uids, etc. The implications are that a realist wouldbe unable to explain the past successes of the theories employing such con-cepts, and that the success of currently accepted theories should not betaken as indicative that their central explanatory concepts are referential,or even approximately referential. But, according to restricted realism,these implications are false; it is not only the instrumental ef�cacy or theempirical adequacy of the discarded non-referential beliefs that can survivein some form, for there is a part of, or an aspect of, what is expressed inthose beliefs that is carried over into the new beliefs. Philip Kitcher haspointed out that the success of a theory, like its con�rmation, is selectivelydistributed. So if a theory is successful on account of the truth of its predic-tions then only those parts of the theory which are required to generatethose predictions can count as contributing to that success, and only thoseparts of the theory can therefore be accorded a realist interpretation. ‘Nosensible realist’, he says, ‘should ever want to assert that the idle parts of anindividual practice, past or present, are justi�ed by the success of the whole’.Accordingly, ‘it is not enough to conceive a theory as a set of statementsand distribute the success of the whole uniformly over the parts. One has tosee how the statements are used’.7

In the case of ‘structural’ scienti�c realism,8 the central idea is thatscienti�c theories do indeed provide information unavailable to us inobservation and experimentation, but that information is about the form orstructure, rather than the nature or content, of what is unobservable. Often,it is claimed, when one theory is replaced by another, it is information

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 73

7 P. Kitcher, The Advancement of Science: Science Without Legend, Objectivity WithoutIllusions (Oxford, Oxford University Press, 1993) pp. 142–3. Cf. C. Hempel, Aspects of Scien-ti�c Explanation (New York, Free Press, 1965) ch.1; C. Glymour, Theory and Evidence(Princeton, NJ., Princeton University Press, 1980) pp. 30–48.

8 Modern interest in structural realism dates from G. Maxwell, ‘Structural realism and themeaning of theoretical terms’, in M. Radner and S. Winokur (eds), Analyses of Theories andMethods of Physics and Psychology, Minnesota Studies in the Philosophy of Science, vol.4,(Minneapolis, Minnesota University Press, 1970), pp. 181–92. Maxwell used the device of‘Ramsey sentences’ to develop a version of structural scienti�c realism. See also G. Maxwell,‘Theories, perception and structural realism’, in R. Colodny (ed.), The Nature and Functionof Scienti�c Theories, (Pittsburgh, Pittsburgh University Press, 1970), pp. 3–34. Russell’sadvocacy of structural realism is discussed in W. Demopoulos and M. Friedman, ‘Criticalnotice: Bertrand Russell’s The Analysis of Matter: its historical context and contemporaryinterest’, Philosophy of Science 52 (1985): 621–39, and in W. Demopoulos and M. Friedman,‘The concept of structure in The Analysis of Matter’, in C. Wade Savage and C. A. Ander-son (eds) Rereading Russell: Essays on Bertrand Russell’s Metaphysics and Epistemology,Minnesota Studies in the Philosophy of Science, vol.12, (Minneapolis, Minnesota UniversityPress, 1989), pp. 183–99.

about the essential nature of what is unobservable that is replaced, ratherthan information about the structure of the unobservable. Consequently,according to the structural scienti�c realist, we can and should avoid theinference from the failure of scienti�c theories in the past to an anti-realistconclusion. We should restrict our scienti�c realism to the claim that know-ledge of what is theoretical in science is con�ned to its structural charac-teristics; we have no knowledge of its intrinsic nature. This distinction needsclari�cation if we are to use it to explain structural realism. What is the basisof the difference between essential and structural properties of, say, a theor-etical entity? And how far is that difference a re�ection of interests andconcerns that may change? No current discussions address these questionsin a direct and explicit manner.9 However, the idea of structural realism,broadly construed, was considered and adopted by a number of philoso-phers in the early decades of this century. We can �nd in Poincaré and inDuhem, in Cassirer, in Schlick and in Carnap, and in Russell, indicationsthat this idea and the questions it raises were examined in a careful andilluminating manner.

The general context for that examination suggested that the ontologicalimport of a theory, whether scienti�c or not, derives from its structurealone.10 Mathematics comes within the scope of this broad context, and it

74 BARRY GOWER

9 There is, though, an analysis of some of the ambiguities in the idea of structuralism as appliedto mathematics in C. S. Chihara, Constructibility and Mathematical Existence (Oxford,Clarendon Press, 1990) ch. 7, and in M. Dummett, Frege: Philosophy of Mathematics(London, Duckworth, 1991) pp. 295–7. The implications of those ambiguities for structuralscienti�c realism await examination. Some will be apparent because of the close connectionsbetween mathematics and mathematical physics, and it is no accident that the examples usedto explain and illustrate structural scienti�c realism are provided by mathematical physics.But it is not obvious that all theories in physics lend themselves to a distinction between‘content’ and ‘structure’. How could theories of atomic structure remain theories about thestructure of atoms if they were not, in fact, about atoms? And if we look to theories otherthan those of physics, the prospects look even less promising. How could we make a dis-tinction between the ‘content’ and the ‘structure’ of Harvey’s theory of the circulation of theblood or of Darwin’s theory of natural selection? There are, of course, some issues herewhich connect with the role of models in science.

10 Bertrand Russell, in his Introduction to Mathematical Philosophy (London, George Allen &Unwin, 1919) wrote: ‘We know much more (to use, for a moment, an old-fashioned pair ofterms) about the form of nature than about the matter. Accordingly, what we really knowwhen we enunciate a law of nature is only that there is probably some interpretation of ourterms which will make the law approximately true’ (p. 55). And, a few pages later, ‘whatmatters in mathematics, and to a very great extent in physical science, is not the intrinsicnature of our terms, but the logical nature of their interrelations’ (p. 59). These claims werenot properly developed until Russell came to write his Analysis of Matter (London, GeorgeAllen & Unwin, 1927). Cf. M. Hallett, ‘Putnam and the Skolem Paradox’, in Peter Clark andBob Hale (eds) Reading Putnam (Oxford, Blackwell, 1994), p. 89: ‘The core of [Russell’s]attitude consists precisely in a shift away from the view that mathematical theories are con-cerned essentially with the description of an independent reality, and towards the view thatthey are at root concerned, not with speci�cations in Poincaré’s sense, but rather withtractable axiom systems.’

is there that we �nd some of the most detailed and sustained attempts toelucidate a distinction between structure and content. Indeed the charac-ter of structural realism, in its widest sense, can be said to have emergedfrom work in the analytic foundations of mathematics by Dedekind andothers at the end of the nineteenth century.11 Any collection of objects canform a structure, and when they do so they constitute the domain of thestructure and will stand in de�nite relations to each other. So far as thestructure is concerned, any object can be replaced by another, providedonly that its relationships are preserved. It is, as it were, the function of anobject within a structure, rather than what is outside that structure, thatmatters. Given this, our knowledge of the objects within the domain of thestructure is con�ned to what we can learn from the ways those objects arerelated to each other. If the objects have characteristics over and abovethese features then they are unknown to us and they play no part in thetheory.12 In a scienti�c theory, the theoretical entities invoked form apattern or structure. The entities, that is to say, are related to each otherin ways that are speci�ed by the mathematical equations of the theory. Noparticular physical system need be determined by a scienti�c theory, sounderstood, and it does not, therefore, provide information about anyphysical system in which the structure is realized. But the theory can never-theless be used to provide testable predictions, for we can use experimentto realize a particular system corresponding to the data available and

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 75

11 See C. Parsons, ‘A structuralist view of mathematical objects’, in W. D. Hart (ed.) The Phil-osophy of Mathematics (Oxford, Oxford University Press, 1996), pp. 275–81, and S. Shapiro,Philosophy of Mathematics: Structure and Ontology (Oxford, Oxford University Press, 1997)pp. 170–6.

12 For an account of how this idea applies in the case of mathematics see M. D. Resnik, ‘Math-ematics as a science of patterns: ontology and reference’, Noûs 15 (1981): 529–50; C. Parsons,‘A structuralist view of mathematical objects’, in W. D. Hart (ed.), The Philosophy of Math-ematics (Oxford, Oxford University Press, 1996); S. Shapiro, Philosophy of Mathematics:Structure and Ontology (Oxford, Oxford University Press, 1997). In M. D. Resnik, Mathe-matics as a Science of Patterns (Oxford, Clarendon Press, 1997) p. 201, the idea is expressedthus:

1 in mathematics the primary subject-matter is not the individual mathematical objectsbut rather the structures in which they are arranged. The objects of mathematics, thatis, the entities which our mathematical constants and quanti�ers denote, are themselvesatoms, structureless points, or positions in structures. And as such they have no iden-tity or distinguishing features outside a structure.

1 G. Hellman in his ‘Modal-structural mathematics’, in A. D. Irvine (ed.), Physicalism in Math-ematics (Dordrecht, Kluwer Academic Publishers, 1990) p. 309, explains that:

1 Broadly speaking, structuralism is the view that mathematical theories typically investi-gate relations holding among items of structures of a given type in abstraction from theidentity of those individual items. As it stands, of course, this is vague: for what countsas a ‘structure’? How are structures characterized? What are the commitments of struc-turalism concerning the intelligibility of higher-order concepts and ontology (e.g. dostructures have to be recognized as actually existing, etc.)?

predict the behaviour of that system using the mathematical equationsexpressing the structure of the theory. If subsequent experimental evidenceshows that the predictions are correct, then we can conclude with an appro-priate degree of con�dence that the theory is correct in what it says aboutform or structure.13 Structural realism invites us, that is, to distinguishbetween what a scienti�c theory is, implicitly or explicitly, about, and whatit says concerning what it is about. If we regard mathematical equations asexpressing structural relationships between phenomena, however theymay be realized, then according to this view scienti�c theories, or at leastthose expressed in mathematical terms, provide us with evidence-tran-scending information about those structural relationships, and not aboutthe way they are realized. Heinrich Hertz expressed the core of this realismwith his well-known declaration: ‘To this question, “What is Maxwell’stheory?” I cannot give any clearer or briefer answer than the following:“Maxwell’s theory is the system of Maxwell’s equations” ’.14 Often, the con-clusions scientists reach about structural relationships do not change overa period of time even though there may be many changes of view aboutwhat these relationships relate. In those circumstances, we would be justi-�ed in believing that the conclusions provide information about the struc-ture or form of reality, though we would not be justi�ed in believing thatthey inform us about the nature or content of the way in which the struc-ture is realized.

More than a century ago, in 1888, Richard Dedekind presented a famousaccount of natural numbers which used, he claimed, only logical notions.What he did was to de�ne a certain ‘simply in�nite system’, and to postu-late natural numbers as ‘free creations of the human mind’ realized in thesystem.15 In seeking an account of arithmetic which would make it con-tinuous with logic and thereby ‘independent of notions or intuitions of spaceor time’, he was sharing common ground with Frege and rejecting the con-structivist epistemology of Kant’s Transcendental Aesthetic. But whereasFrege regarded numbers as having a mind-independent nature or essencewhich enables us to say what it is about them that enables them to satisfythe system he speci�es, Dedekind’s view that they are ‘free creations’

76 BARRY GOWER

13 According to a recent account of the semantic conception of scienti�c theories in F. Suppe,The Semantic Conception of Theories and Scienti�c Realism (Urbana, Illinois UniversityPress, 1989) p. 84: ‘we analyse theories as relational systems consisting of a domain con-taining all (logically) possible states of all (logically) possible physical systems for the theorytogether with various attributes de�ned over that domain’.

14 H. Hertz, Electric Waves: Being Researches on the Propagation of Electric Action with FiniteVelocity Through Space, translated by D. E. Jones (New York, Dover Publications, 1962),p. 21. First published in 1892.

15 R. Dedekind, Was sind und was sollen die Zahlen?, translated by W. W. Beman, revised byW. B. Ewald, in W. B. Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundationsof Mathematics, vol.2 (Oxford, Clarendon Press, 1996) §73. Originally published in 1888.

implies a rejection of that platonistic view as well as a rejection of a con-structivist picture of arithmetic incorporating constraints deriving fromhuman capacities to imagine or to construct. To this extent, Dedekind doesnot tell us what natural numbers are. He invites, indeed, a structuralistinterpretation of his account when he says:

If in the consideration of a simply in�nite system N ordered by a mapping f weentirely neglect the special character of the elements, simply retaining their dis-tinguishability and taking into account only the relations to one another inwhich they are placed by the mapping f , then these elements are called naturalnumbers or ordinal numbers or simply numbers.16

Certainly Bertrand Russell appears to interpret Dedekind’s theory as anexpression of structuralism when he complains that:

it is impossible that the ordinals should be, as Dedekind suggests, nothing butthe terms of such relations as constitute a progression. If they are anything atall, they must be intrinsically something; they must differ from other entities aspoints from instants, or colours from sounds.17

But this complaint ignores Dedekind’s implicit acknowledgement that theelements of his system, the natural numbers, do have a ‘special character’which must derive from numbers being ‘free creations’. Numbers are, itseems, to be understood as particular mental things, and the question thatthis understanding raises is how, then, can we explain the objectivity ofarithmetic. Although Dedekind does not address this question explicitly, he

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 77

16 Dedekind, op.cit. (note 15) p. 809. According to Stein, Dedekind was ‘ontologically indif-ferent’ about natural numbers because he thought that ‘it does not matter what numbersare; what matters is that they constitute a simply in�nite system’ (H. Stein, ‘Logos, logic, andLogistiké: some philosophical remarks on the nineteenth century transformation of mathe-matics’, in W. Aspray and P. Kitcher (eds), History and Philosophy of Modern Mathemat-ics, Minnesota Studies in the Philosophy of Science, vol. 11 (Minneapolis, MinnesotaUniversity Press, 1988) p. 247). This is to attribute to Dedekind what has been called an‘eliminative’ version of structuralism, according to which statements apparently about a kindof mathematical object, such as numbers, are to be understood as general statements aboutspeci�ed structures enabling us to eliminate reference to mathematical objects of the kindin question (see C. Parsons, ‘A structuralist view of mathematical objects’, in W. D. Hart(ed.) The Philosophy of Mathematics (Oxford, Oxford University Press, 1996) p. 277).Stein’s view is challenged by McCarty on the grounds that it fails to take account ofDedekind’s repeated and consistently held claim that numbers are ‘free creations of thehuman mind’ (C. McCarty, ‘The mysteries of Richard Dedekind’, in J. Hintikka (ed.), FromDedekind to Gödel (Dordrecht, Kluwer Academic Publishers, 1995) pp. 68–70). For dis-cussion of the issue, see S. Shapiro, Philosophy of Mathematics: Structure and Ontology(Oxford, Oxford University Press, 1997) pp. 172–6.

17 B. Russell, The Principles of Mathematics, second edition (London, George Allen & Unwin,1937) p. 249. The �rst edition was published in 1903.

seems to have believed that, in Kantian terms, it is our reason that createsnumbers, and that the logical constraints on reason imposed by the need fornumbers to satisfy his simply in�nite system ensure that what one personcreates will coincide in its arithmetical properties with what another personconstructs.18 Numbers are mental particulars but nevertheless our know-ledge of numerical truths is objective because numbers are free creations ofhuman reason and can have no properties other than the structural or rela-tional properties determined by their role in Dedekind’s system. There maywell be fundamental disagreements about the subject matter or content ofarithmetic even though this does not result in disputes about arithmeticaltruths. Thus, Frege observed that, in a quite literal sense, mathematiciansdo not know what they are talking about. For some, arithmetical claims areabout abstract particulars called ‘numbers’; for others, such claims are aboutsets; for still others they are about mental entities; for yet others they areabout marks on paper or vocal sounds:

This resembles what it would be like if botanists were not agreed about whatthey wished to understand by a plant, so that for one botanist a plant was, say,an organically developing structure, for another a human artefact, and for athird something that was not perceptible by the senses at all.19

We do not need to decide whether Dedekind was or was not a mathemat-ical structuralist. It is sufficient to notice that some elements of argumentsand claims of structuralists can be found in his writings, and to notice thecontext for those arguments and claims. Nor do we need to explore therelationship between Dedekind’s account of natural numbers andHilbert’s treatment of geometry, though it is plain that both rejected theKantian epistemology. For Hilbert in his ‘Grundlagen der Geometrie’ thegeometrical concepts of point and line are implicitly defined as whateversatisfy the axioms of geometry. Intuitive ideas of what are points and linescan be set aside as psychological irrelevancies. ‘It must always be possible’,Hilbert reportedly said, ‘to replace [in geometric statements] the words“points”, “lines”, “planes”, by “tables”, “chairs”, “mugs”.’ As Frege cor-

78 BARRY GOWER

18 See C. McCarty, ‘The mysteries of Richard Dedekind’, in J. Hintikka (ed.) From Dedekindto Gödel (Dordrecht, Kluwer Academic Publishers, 1995) pp. 69, 82–3.

19 G. Frege, ‘Logic in mathematics’, in G. Frege, Posthumous Writings, translated by P. Longand R. White (Oxford, Basil Blackwell, 1979), p. 215. For discussion of Frege’s objectionsto Dedekind’s ‘structuralist’ account of numbers, see M. Dummett, Frege: Philosophy ofMathematics (London, Duckworth, 1991) ch.5, and W. W. Tait, ‘Frege versus Cantor andDedekind: on the concept of number’, in W. W. Tait (ed.) Early Analytic Philosophy: Frege,Russell, Wittgenstein: Essays in Honor of Leonard Linsky (Chicago and La Salle, Illinois,Open Court, 1997) pp. 213–48. See also H. Poincaré, Science and Method, translated byFrancis Maitland (London, Thomas Nelson and Sons, 1914) p. 155: ‘The de�nitions ofnumber are very numerous and of great variety.

rectly observed of Hilbert’s approach: ‘you want to detach geometryentirely from spatial intuitions’.20 Hilbert’s axiomatization of geometrydoes not result in statements which describe a subject matter revealed byintuition, but in a relational structure which fixes and exhausts themeaning of the language in which the structure is described.21 That is tosay, meanings can be given, in the form of ‘implicit definitions’, by therelationships themselves. The traditional view was that we need to knowwhat numbers are, or what geometrical distance is, or what atoms are, (orwhat the words ‘seven’, ‘distance’ or ‘atom’ mean) before we can decidewhat statements about numbers, distance, or atoms are true.22 This view

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 79

20 D. Hilbert, Gesammelte Abhandlungen, 3Bd. (Berlin, Julius Springer Verlag, 1935) p. 403;cf. J. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station,edited by L. Wessells (Cambridge, Cambridge University Press, 1991) p. 135, and S. Shapiro,Philosophy of Mathematics: Structure and Ontology (Oxford, Oxford University Press, 1997)p. 157. G. Frege, Philosophical and Mathematical Correspondence, translated by Hans Kaal(Chicago, Chicago University Press, 1980) p. 43. See also M. Hallett, ‘Physicalism, reduc-tionism and Hilbert’ in A. D. Irvine (ed.) Physicalism in Mathematics (Dordrecht, KluwerAcademic Publishers, 1990) p. 196: ‘According to Frege, axioms should be self-evidenttruths. Thus, it should be clear what they are about and that they are true of this’. In a letterto Frege, Hilbert says:

1 I do not want to assume anything as known in advance; I regard my explanation insect.1 [of ‘Grundlagen der Geometrie’] as the definition of the concepts point, line,plane . . . If one is looking for other definitions of a ‘point’, e.g. through paraphrasein terms of extensionless, etc., then I must indeed oppose such attempts in the mostdecisive way; one is looking for something one can never find because there is nothingthere; and everything gets lost and becomes vague and tangled and degenerates intoa game of hide-and-seek.

1 (G. Frege, Philosophical and Mathematical Correspondence, translated byHans Kaal (Chicago, Chicago University Press, 1980) p. 39)

21 See, for example, Paul Bernay’s article ‘Hilbert, David’ in P. Edwards (ed.) The Encyclo-pedia of Philosophy (New York, Macmillan and The Free Press, 1967) vol. 3, pp. 496–504.

22 Frege, in a letter to Hilbert, expressed this traditional view as follows: ‘Thus, axioms andtheorems can never try to lay down the meaning [Bedeutung] of a sign or word that occursin them, but it must be already laid down’ (G. Frege, Philosophical and Mathematical Cor-respondence, translated by Hans Kaal (Chicago, Chicago University Press, 1980) p. 36). Itis, indeed, because meanings are predetermined that axioms can be ‘self-evidently’ true.Hilbert, in his contribution to the correspondence, expressed his opposition to this tra-ditional view in this way:

1 You say that my concepts, e.g. ‘point’, ‘between’, are not unequivocally �xed . . . But it issurely obvious that every theory is only a scaffolding or schema of concepts together withtheir necessary relations to one another, and that the basic elements can be thought of inany way one likes. . . In other words: any theory can always be applied to in�nitely manysystems of basic elements. . . All the statements of the theory of electricity are of coursealso valid for any other system of things which is substituted for the concepts magnetism,electricity . . . provided only that the requisite axioms are satis�ed. But the circumstancesI mentioned can never be a defect in a theory, and it is in any case unavoidable.

(G. Frege, Philosophical and Mathematical Correspondence, translated byHans Kaal (Chicago, Chicago University Press, 1980) p. 41)

was challenged at the end of the nineteenth century: our decisions aboutwhat statements are true determine, or more usually underdetermine,what numbers, etc. are. We have, since then, become familiar with the ideathat the formalism of a theory can be given alternative semantic interpre-tations, and that sometimes the intended interpretation can be modeled byan alternative which makes the claims of the theory true. Mathematicalstructures can qualify as mathematical models, and the claim of structuralrealism is that when they do the realist’s demand for evidence-transcend-ing information can be satisfied.23

2

The view we know as structural scienti�c realism was explicitly and clearlyexpressed by Poincaré.24 He claimed that the reality revealed by scienceconcerns the relations between objects, not the objects themselves:

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23 The structuralism of Dedekind and Hilbert might be thought to raise familiar issues aboutthe models that exemplify a structure, and therefore about the interpretation of the state-ments specifying the structure. Do we have any assurance that nothing crucial turns on thechoice of a model, and that the interpretation indicated by the choice of a model is intended?In Dedekind’s case the ‘free creations’ that are the natural numbers are abstractions whichcan be realized, or made concrete, in different models; but these models, he claimed, are iso-morphic and therefore equivalent from the point of view of the system they exemplify. Butthe Löwenheim–Skolem theorem claims that non-standard models of a structure with an in�-nite model cannot be ruled out. So there are, possibly, non-standard models and unintendedinterpretations of Dedekind’s simply in�nite system. But although this purely formal resultmight show that realism about the system in so far as it is displayed in a model, or in a set ofisomorphic models, is mistaken, it does not show that realism about the structural relationsof the system is also mistaken. In Hilbert’s case, we do not need to, and should not, �x thereference of the terms used to specify the structure of Euclidean geometry. So the prospectof non-standard models and unintended interpretations can be set aside as irrelevant. It is theform or structure of Euclidean geometry which matters to Hilbert, and it is this form or struc-ture which is �xed and real. The ‘content’ of Euclidean geometry – what it is about – is not�xed and realists can envisage non-standard interpretations or content with equanimity.

24 H. Poincaré, Science and Hypothesis, translated by W. J. Greenstreet (New York, DoverPublications, 1952) p. 20:

1 Mathematicians do not study objects, but the relations between objects; to them it is amatter of indifference if these objects are replaced by others, provided that the relationsdo not change. Matter does not engage their attention, they are interested in form alone.

1 For modern discussion of Poincaré’s views, see J. Worrall, ‘Structural realism: the best of bothworlds?’, in D. Papineau (ed.) The Philosophy of Science (Oxford, Oxford University Press,1996) pp. 139–65; J. Worrall, ‘How to remain (reasonably) optimistic: scienti�c realism andthe “luminiferous ether” ’, in D. Hull, M. Forbes and R. M. Burian (eds) PSA 1994, vol.1, (EastLancing, Michigan, Philosophy of Science Association, 1994), pp. 334–42; S. Psillos, ‘Is struc-tural realism the best of both worlds?’ Dialectica 49 (1995): 15–46; E. Zahar, ‘Poincaré’s struc-tural realism and his logic of discovery’, in J. L. Greffe, G. Heinzmann and K. Lorenz (eds)Henri Poincaré: Science and Philosophy, International Congress Nancy, France, 1994 (Parisand Berlin, A. Blanchard and Akademie Verlag, 1996) pp. 45–68.

real objects . . . Nature will hide forever from our eyes, [and] the true relationsbetween these real objects are the only reality we can attain, and the solecondition is that the same relations shall exist between these objects as betweenthe images we are forced to put in their place.25

In the light of the remarkable systematic success of theories in the math-ematical sciences, we cannot accept an anti-realist view of them. Theycannot, Poincaré claimed, be ‘simple practical recipes’ for predictions, forthat would oblige us to attribute their success to chance. But nor can weaccept a straightforward realist view of these theories, for although manyof them are successful over a period of time, the way in which they changemakes it implausible to suppose that they provide ever closer approxi-mations to the whole truth. The history of science, he said, shows how‘ephemeral’ are theories in the physical science. But still, he continued,‘they do not entirely perish, and of each of them some traces still remain’.The task is to discover the nature of these traces ‘because in them andin them alone is the true reality’.26 And Poincaré’s conclusion was thatthe mathematical equations of physical science contain the traces in ques-tion, for they ‘express relations, and if the equations remain true, it isbecause the relations preserve their reality’. What matters, he says, is thattheories succeed in capturing ‘the true relations between . . . real objects’,27

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 81

25 H. Poincaré, op.cit. (note 24) p. 161. The same point is made in the ‘Author’s Preface’ toScience and Hypothesis, p.xxiv: ‘the aim of science is not things themselves, as the dogma-tists in their simplicity imagine, but the relations between things; outside those relationsthere is no reality knowable’.

26 H. Poincaré, op.cit. (note 24) p.xxvi. Cf. Poincaré’s remarks in his The Value of Science, trans-lated by George B. Halsted, in H. Poincaré, The Foundations of Science: Science and Hypoth-esis, The Value of Science, and Science and Method (Lancaster. Pa., The Science Press, 1913)p. 351:

1 science has already lived long enough for us to be able to �nd out by asking its historywhether the edi�ces it builds stand the test of time, or whether they are only ephemeralconstructions. Now what do we see? At �rst blush it seems to us that the theories lastonly a day and that ruins upon ruins accumulate. To-day the theories are born, to-morrow they are the fashion, the day after to-morrow they are classic, the fourth daythey are superannuated, and the �fth they are forgotten. But if we look more closely,we see that what thus succumb are the theories properly so-called, those that pretendto teach us what things are. But there is in them something which usually survives. Ifone of them taught us a true relation, this relation is de�nitively acquired, and it willbe found again under a new disguise in the other theories which will successively cometo reign in place of the old.

1 And he concludes, p. 352: ‘the sole objective reality consists in the relations of things . . .Doubtless these relations . . . could not be conceived outside of a mind which conceivesthem. But they are nevertheless objective because they are, will become, or will remain,common to all thinking beings’.

27 Poincaré, op.cit. (note 24) p. 161.

despite the fact that the real objects in question are inaccessible to us.In thought, we use images to designate these inaccessible real objects, andwe sometimes find it ‘convenient’ to replace one image by another. But if,as sometimes happens, this replacement has no effect on our claims aboutthe relations between the real objects designated by our images thendespite the replacement our understanding of reality will not havechanged. Therefore, as Poincaré argued in a preface he wrote to his texton theoretical optics, ‘Mathematical theories do not have as their objectto reveal to us the true nature of things; that would be an unreasonableaspiration’.28

Looked at from our vantage point at the end of the twentieth century,and in the light of our preoccupations, it does indeed look as though Poin-caré was trying to find a compromise between the realism that invites uswhen we consider the so-called ‘no miracles’ argument, and the anti-realism that is the conclusion of the so-called ‘pessimistic meta-induction’.Put another way, structural scientific realism enables us to have the ‘bestof both worlds’. That may be so, but we must be careful not to misrepre-sent Poincaré. He does, it is true, mention in support of his version ofrestricted realism just those arguments which preoccupy us, but neverthe-less his views are inevitably grounded in the nineteenth, not the twentieth,century. Unfortunately, Poincaré did not set out his philosophical views ina systematic manner so that we can see how they are so grounded, and itis not at all easy to reconstruct them with confidence. However, it is clearthat his well-known geometric conventionalism was in part a response tocomplaints, following the development of non-Euclidean geometries,about Kant’s pure intuition. Geometrical axioms are neither analytic norsynthetic, and they are not truths which are known either a priori or empir-ically. They are, rather, ‘definitions in disguise’ which we may choose touse or not. Kant’s view was that we know, by pure intuition, what geo-metrical primitives such as points, lines and planes are, and geometricalaxioms state a priori truths about these primitive notions; Poincaré’s viewis that everything we know about points, lines and planes we know byvirtue of our decision to adopt certain axioms as ‘disguised’ definitions ofthese notions. In the context of his debate with Russell at the very end ofthe nineteenth century, it became clear that ‘disguised’ definitions serve toidentify an object in terms of the relation or relations in which it stands toother objects. So, for example, we give a ‘disguised’ definition of the objectC if we define it as the (unique) object lying between objects A and B. Such

82 BARRY GOWER

28 Quoted in P. Duhem, Essays in the History and Philosophy of Science, translated and edited,with an introduction, by R. Ariew and P. Barker, (Indianapolis and Cambridge, HackettPublishing Co., 1996) p. 19 from the ‘Préface’ to H. Poincaré, Théorie de mathématique dela lumière.

a definition defines an object by specifying its role in a system or patternor structure. But we might think, and Russell did think, that there isanother sense of definition according to which we still lack knowledge ofan object even when we know the role it has in a structure. We still needa definition of C even though we know it is the unique object lying betweenA and B. In case the object should be an unanalysable primitive and there-fore undefinable in this second sense, it will be known, if known at all,directly through intuition, or ‘acquaintance’ as Russell was later to say.The idea that motivates this second sense of definition has been called the‘thesis of semantic atomism’: it says that ‘if a sentence S is to convey infor-mation . . . , then its grammatical units must have a meaning before theyjoin their partners in S’.29 In Poincaré’s view, the lesson to be learned fromthe advent of non-Euclidean geometry was that the grammatical units usedin statements expressing geometrical axioms do not and cannot have a pre-determined meaning, and that we know nothing about geometrical enti-ties independently of the axioms themselves. It follows from the thesis ofsemantic atomism that we cannot understand the sentences expressinggeometrical axioms as conveying information; they do not in particularconvey information about geometric entities; they are ‘disguised’ defi-nitions and as such have a conventional rather than a propositional char-acter. For Poincaré, geometric axioms tell us everything we can knowabout the meanings of primitive geometrical concepts.30

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 83

29 J. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station,edited by L. Wessells (Cambridge, Cambridge University Press, 1991) p. 131. Zahar claimsthat one difficulty for structural realism is that ‘no coherent semantics for this approach isas yet available’ (E. Zahar, ‘Poincaré’s structural realism and his logic of discovery’, in J.L. Greffe, G. Heinzmann and K. Lorenz (eds), Henri Poincaré: Science and Philosophy,International Congress Nancy, France, 1994 (Paris and Berlin, A. Blanchard and AkademieVerlag, 1996) p. 47). This is because, according to traditional semantics, i.e. semanticatomism, we cannot properly understand any statement if we do not know what that state-ment is about. In particular, traditional semantics can only give meaning to relational state-ments by giving meaning to relata, i.e. the terms denoting what is related.

30 For Poincaré’s endorsement of Hilbert’s approach to geometry as expressed in ‘Grundlagender Geometrie’ see his Science and Method, translated by F. Maitland (London, ThomasNelson and Sons, 1914) p. 147:

1 What strikes us . . . in the new mathematics is its purely formal character: ‘Imagine’,says Hilbert, ‘three kinds of things, which we will call points, straight lines, andplanes. . . .’ What these things are, not only we do not know, but we must not seek toknow. It is unnecessary, and one who had never seen either a point or a straight line ora plane could do geometry just as well as we can. . . Thus it will be readily understoodthat, in order to demonstrate a theorem, it is not necessary or even useful to know whatit means. We might replace geometry by the reasoning piano imagined by StanleyJevons. . . It is no more necessary for the mathematician than it is for these machinesto know what he is doing.

1 The quotation from D. Hilbert can be found in his The Foundations of Geometry, translatedby E. J. Townsend (Chicago, Open Court Publishing Co., 1902) p. 3. See also M. Hallett,

If we translate these ideas into the context of mathematical physics wesee an additional, and deeper, source for structural realism. We do not knowabout theoretical entities independently of the theoretical principles con-cerning them; theoretical terms do not have a meaning prior to that giventhem by their role in theoretical principles. We might think that we do havesuch knowledge when we try to picture, with the help of a model, what atheory says. But for Poincaré this would be to embrace anthropomorphism:if such a person:

claims that all physics can be explained by the mutual impact of atoms [and]simply means that the same relations obtain between physical phenomena asbetween the mutual impact of a large number of billiard balls – well and good!This is veri�able and perhaps true. But he means something more, and we thinkwe understand him because we think we know what an impact is. Why? Simplybecause we have often watched a game of billiards. Are we to understand thatGod experiences the same sensations in the contemplation of his work that wedo in watching a game of billiards? If it is not our intention to give his assertion

84 BARRY GOWER

‘Physicalism, reductionism and Hilbert’, in A. D. Irvine (ed.) Physicalism in Mathematics,(Dordrecht, Kluwer Academic Publishers, 1990) pp. 199–200:

1 What kind of direct insight could we possibly have into the notion of ‘point’ or ‘line’?The dif�culties here are apparent from the highly unclear attempts made by Euclid toexplain the basic geometrical terms at the beginning of the Elements. One of the waysin which Hilbert’s work on geometry was an improvement over the system of Euclid isthat it dispenses with such explanations (and in particular with extensive reliance onintuition as a source of direct knowledge of the objects of the science), and the natureof the improvement, if good in this case, is surely quite general. . . . Hilbert claimed, ineffect, that we do not need any kind of direct or extra insight into the referents of thebasic terms of mathematics in order to be able to understand mathematical theories:the axioms governing the concepts involved contain all the insight we get or need.According to Hilbert, the axioms of geometry or real numbers are hence more like col-lective ‘implicit de�nitions’ of the terms that �gure in them than ‘basic truths’ in Frege’ssense. . . . Hilbert clearly intends this notion of de�nition in a strong sense, namely thatthe axiom system completely �xes the meaning of the concepts involved.

1 A further aspect of the general background for what Poincaré says concerning structuralrealism in science is to be found in his discussion of objectivity in science in The Value ofScience. There, after drawing attention to the incommunicability (‘intransmissibility’) of thecontent of sensations and the communicability of relations between sensations, he says:‘nothing is objective which is not transmissible, and consequently . . . the relations betweenthe sensations can alone have an objective value’ and ‘science . . . is a system of relations[and] it is in the relations alone that objectivity must be sought’. The ‘objective value ofscience’ he says, does not lie in the ability of science to ‘teach us the true nature of things’,for neither it nor any other kind of enquiry has such an ability; it lies, rather, in the abilityof science to ‘teach the true relations of things . . . If any god knew [the nature of things],he could not �nd words to express it [and] I ask myself even whether we really understandthe question’ (H. Poincaré, The Value of Science, translated by G. B. Halsted, in H. Poin-caré, The Foundations of Science: Science and Hypothesis, The Value of Science, and Scienceand Method (Lancaster. Pa., The Science Press, 1913) pp. 348–50).

this fantastic meaning, and if we do not wish to give it the more restrictedmeaning I have already mentioned, which is the sound meaning, then it has nomeaning at all.31

The attempts of the physicist to go beyond knowledge of structure asexpressed in theoretical statements about relations between physicalphenomena, and acquire knowledge of the phenomena themselves is notonly unnecessary, but founded on questionable philosophy. Kant hadclaimed that concepts without intuitions are empty, but in mathematics andthe mathematical sciences it was becoming clear how this claim could bechallenged. Misgivings about the necessity or indeed legitimacy of the roleof intuition in geometry lay behind and motivated Poincaré’s conventional-ism; and misgivings about the necessity or legitimacy of pictures or modelsintended to convey information, albeit by analogy, about the subject matterof theoretical principles in mathematical science led to his structuralistaccount of those principles. For if the arrival of non-Euclidean geometrieshad demonstrated the dispensability and indeed the impossibility ofknowing what points, lines, planes are independent of geometrical axioms,the development of mathematical physics in the nineteenth century in thehands of French and German analysts had shown that reason, without theaid of imaginative pictures or models, was entirely suf�cient to meet reason-able demands for knowledge and understanding.

It is of course to Duhem that we look for a sustained critique of ‘English’methods, and their dependence on models. But the well-known chapter inThe Aim and Structure of Physical Theory, which sets out the grounds forhis hostility to those methods, is preceded by two chapters in which we �nda neglected but clear re-statement of Poincaré’s claim that theories revealthe truth about structural relations in nature rather than the truth aboutrelata. ‘Physical theory’, he says, ‘never reveals realities hiding under thesensible appearances; but the more complete it becomes . . . the more wesuspect that the relations it establishes among the data of observation cor-respond to real relations among things’. We cannot appeal to observationfor a proof that this ‘suspicion’ is correct, for observation itself ‘cannotprove that the order established among experimental laws re�ects an ordertranscending experience’. But nevertheless, a physicist ‘cannot . . . believe

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31 H. Poincaré, Science and Hypothesis, translated by W. J. Greenstreet (New York, DoverPublications, 1952) pp. 163–4. See also Poincaré’s remarks, in the special preface he wroteto Halsted’s translations, on the differences between the characteristic approaches of‘Latins’ and ‘Anglo-Saxons’. For example:

The Anglo-Saxon to depict a phenomenon will �rst be engrossed in making a model, andhe will make it with common materials, such as our crude, unaided senses show us them.He also makes a hypothesis, he assumes implicitly that nature, in her �nest elements, isthe same as in the complicated aggregates which alone are within the reach of our senses.He concludes from the body to the atom.

(H. Poincaré, The Foundations of Science: Science and Hypothesis, The Value ofScience, and Science and Method (Lancaster, Pa., The Science Press 1913) p. 6)

that a system capable of ordering so simply and so easily a vast number oflaws, so disparate at �rst encounter, should be a purely arti�cial system’.That is to say, it is impossible for a physicist to believe that the success ofphysical theory is ‘a marvelous feat of chance’; he or she believes, rather,that the success of a theory is a consequence of its being a natural rep-resentation of ‘the real relations among the invisible realities’.32 Theseremarks make it clear that Duhem endorses a central argument for scien-ti�c realism; so he needs to �nd a way of reconciling this endorsement withhis well-known arguments which appear to support a version of scienti�canti-realism. To this end he distinguishes between what he calls the‘explanatory role of a theory’ and the ‘representative role of a theory’. Toful�ll the former – explanatory – role a theory must inform us about ‘invis-ible realities’, and in Duhem’s view, as explained in the �rst chapter of hisbook, this is a task beyond the capacity and scope of any physical theory.To ful�ll the latter – representative – role a theory must inform us aboutrelations, and in Duhem’s view this is a task successful physical theories canand do perform. ‘Everything good in [a] theory’, he says, ‘ by virtue of whichit . . . [has] the power to anticipate experience, is found in the representa-tive part’, whereas ‘whatever is false in [a] theory and contradicted by thefacts is found above all in the explanatory part; the physicist has broughterror into it, led by his desire to take hold of realities’. As a consequence,he says:

When the progress of experimental physics goes counter to a theory andcompels it to be modi�ed or transformed, the purely representative part entersnearly whole in the new theory, bringing to it the inheritance of all the valuablepossessions of the old theory, whereas the explanatory part falls out in order togive way to another explanation.33

For both Poincaré and Duhem, then, a defensible scienti�c realism must bestructural in the sense that it attributes reality to the relational structure ofa scienti�c theory. Given the in�uence that both these thinkers had on thefounders of logical empiricism, it is not surprising that aspects of this themereappear in their work. But we should not expect that the assimilation is atall straightforward. For, in the �rst place, the conventionalisms that Poin-caré and Duhem coupled with their realism differed in important respects.Certainly, there are complexities and ambiguities in the ‘conventionalistrealism’ of Schlick, as we shall see. And in the second place, structural scien-ti�c realism was being developed in a rather different way by anotherthinker whose views had a signi�cant effect on early logical empiricism,Ernst Cassirer.

86 BARRY GOWER

32 P. Duhem, The Aim and Structure of Physical Theory, translated by P. P. Wiener (Prince-ton, NJ., Princeton University Press, 1954) pp. 26–8.

33 Duhem, op.cit. (note 32) p. 32.

3

The �rst edition of Duhem’s Aim and Structure of Physical Theory was pub-lished in 1906. It was in that year that Ernst Cassirer accepted a position atBerlin University where he was to remain as privatdozent until 1919. He wasone among a group of so-called Marburg neo-Kantian philosophers, includ-ing Paul Natorp and Hermann Cohen, working in the �rst two decades of thiscentury. In using the label ‘neo-Kantian’ we need to appreciate that thefundamental contrast for many German philosophers of the late nineteenthand early twentieth century was between the Kantian critical philosophy andthe positivism of, for example, Mach.34 The divergent approaches of thesetwo philosophies to the epistemology of mathematics and the exact scienceswas particularly important. Of the very many ways to express this contrastperhaps the simplest is in terms of attitudes towards synthetic a priori know-ledge. For Kantians, some knowledge is, or is like, synthetic a priori know-ledge; for positivists there is no such knowledge. Bearing in mind that therecan be many different understandings of synthetic a priori knowledge, not allof which would have been recognized, still less accepted, by Kant himself, itis evident that there is considerable scope for the development of differentkinds of neo-Kantianism. For many, though not all, an alternative way ofexpressing the same contrast was in terms of pure intuition: the possibility ofsynthetic a priori knowledge required the possibility of pure intuition.

For Kant and the neo-Kantians, though not for the positivists, the centralphilosophical problem was to �nd a way of securing not the certainty or reli-ability of empirical knowledge but its objectivity, given that our experience

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 87

34 See for example R. Carnap, The Logical Structure of the World, translated by R. George(Berkeley, University of California Press, 1969) §75:

34 Positivism has emphasized that the only material of cognition consists in the undigestedexperientially given. It is here that we have to look for the basic elements of the con-structional system. Transcendental idealism, especially the Neo-Kantian school (Rickert,Cassirer, Bauch), has justly emphasized that these elements do not suf�ce. Order con-cepts, our basic relations, must be added.

1 This in�uential monograph was originally published as Der logische Aufbau der Welt,(Berlin–Schlachtensee, Weltkreis–Verlag, 1928). For an example of the way in which Kantand Mach represent opposite trends, see M. Schlick, General Theory of Knowledge, Libraryof Exact Philosophy XI, translated by A. E. Blumberg (Vienna, Springer–Verlag, 1974, esp.ch.3: ‘Problems of Reality’). This is a translation of the second edition of Allgemeine Erken-ntnislehre (Berlin, Springer, 1925). The �rst edition was published in 1918. Cf. J. AlbertoCoffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station, edited by L. Wes-sells (Cambridge, Cambridge University Press, 1991) p. 1:

1 Within the �eld of epistemology one may discern three major currents of thought inthe nineteenth century: positivism, Kantianism, and what I propose to call the seman-tic tradition. What distinguished their proponents primarily was their attitude to the apriori. Positivists denied it, and Kantians explained it through the Copernican revol-ution. The semantic tradition consisted of those who believed in the a priori but notin the constitutive powers of the mind.

is essentially private and subjective.35 In terms of the distinction betweenform and content, since the content of experience is subjective we must �nda way of using its form to underwrite the objectivity of the knowledge ityields and thereby bridge, as it were, the gap between thought and reality.Put brie�y, if crudely, Kant’s claim was that it is a ground or condition ofthe intelligibility of subjective experience that we contribute to it a formsupplied by pure understanding. There is a part of scienti�c knowledge,therefore, which does not derive from experience, but results from apply-ing to subjective experience what Kant calls the pure concepts of the under-standing. And this part of science – the non-empirical part as it were –makes empirical knowledge possible.36 It consists of synthetic truths knowna priori, and ‘transcendental logic’ is the study of these truths. There are,that is to say, formal or structural features of our experience which are pro-vided by us and which make objective empirical knowledge possible. TheMarburg neo-Kantians set themselves the task of articulating a view of thiskind, taking into account the advances in the exact sciences, including Ein-stein’s new and revolutionary ideas about space, time and gravitation, theintroduction of a more rigorous approach to analysis in mathematics, andthe consequences of the new logic of classes and relations developed byFrege and Russell.37 This last development was particularly importantbecause it was seen as providing the elements of a new transcendental logic.And with the aid of the logically necessary laws of thought supplied by this

88 BARRY GOWER

35 Cf. T. A. Ryckman, ‘Conditio sine qua non? Zuordnung in the early epistemologies of Cas-sirer and Schlick’, Synthese 88 (1991): p. 66: in the ‘method of Erkenntniskritik . . . theconcern is not the justi�cation per se of scienti�c knowledge (for such knowledge is not indispute, at least in the exact sciences), but rather with how the objectivity of this knowledgeis constituted’. For example, Kant in his Prolegomena To Any Future Metaphysics, trans-lated by P. G. Lucas (Manchester, Manchester University Press, 1953), p. 56 said: ‘Empiri-cal judgements, so far as they have objective validity, are judgements of experience. . . .[They] always need . . . special concepts originally generated in the understanding, and it isthese that make the judgements of experience objectively valid.’

36 Cf. M. Friedman, ‘Geometry, convention, and the relativized a priori: Reichenbach, Schlick,and Carnap’, in W. C. Salmon and G. Wolters (eds) Logic, Language, and the Structure ofScienti�c Theories (Pittsburgh and Konstanz, Pittsburgh University Press and Univer-sitätsverlag Konstanz, 1994). Friedman distinguishes between the ‘constitutive principles’ ofthe non-empirical part of science, and the laws of empirical science.

37 P. Natorp’s Die logischen Grundlagen der exacten Wissenschaften, Wissenschaft undHypothese, vol.12 (Leipzig, B. G. Teubner, 1910) contained the �rst important philosophi-cal study of relativity theory. For comments on this study see D. Howard, ‘Einstein and Ein-deutigkeit: a neglected theme in the philosophical background to General Relativity’, in J.Eisenstaedt and A. J. Kox (eds) Studies in the History of General Relativity, Einstein Studies,vol.3 (Boston, Birkhäuser, 1992) pp. 187–92; D. Howard, ‘Einstein, Kant, and the origins oflogical empiricism’, in W. C. Salmon and G. Wolters (eds), Logic, Language, and the Struc-ture of Scienti�c Theories (Pittsburgh and Konstanz, Pittsburgh University Press and Uni-versitätsverlag Konstanz, 1994) pp. 50–1; D. Howard, ‘Relativity, Eindeutigkeit, andmonomorphism: Rudolf Carnap and the development of the categoricity concept in formalsemantics’, in R. N. Giere and A. W. Richardson (eds) Origins of Logical Positivism, Min-nesota Studies in the Philosophy of Science, vol.16 (Minneapolis, Minnesota UniversityPress, 1996) pp. 133–5.

logic we can, in the exact sciences at least, transform subjective experienceinto objective knowledge of reality.38

Cassirer’s principal contribution to Marburg neo-Kantianism was hisbook Substance and Function, �rst published in 1910. The overall aim of thearguments advanced in this book is that of dispensing with concepts of sub-stance in science in favour of concepts of relations or functions.39 Thosearguments build upon the logicism of Russell and others in showing, as Cas-sirer puts it, that ‘mathematics is representable as nothing other than aspecial application of the general logic of relations; however, the relationconcept in turn goes back to the more fundamental idea of “functional-ity” ’.40 Instead of presupposing, in accordance with the emphasis on

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38 See M. Friedman, ‘Epistemology in the Aufbau’, Synthese 93 (1992): pp. 22–4, for a brief buthelpful characterization of neo-Kantian themes. According to Cassirer transcendental logicis still synthetic; it receives its justi�cation through its grounding of the objectivity of empiri-cal science – see W. Sauer, ‘On the Kantian background of neopositivism’, Topoi 8 (1989):116; A. W. Richardson, ‘Logical idealism and Carnap’s construction of the world’, Synthese93 (1992): 66; and T. A. Ryckman, ‘Conditio sine qua non? Zuordnung in the early episte-mologies of Cassirer and Schlick’, Synthese 88 (1991): 62.

39 For further explanation of this important aspect of Cassirer’s Substance and Function, seeT. A. Ryckman, ‘Conditio sine qua non? Zuordnung in the early epistemologies of Cassirerand Schlick’, Synthese 88 (1991): 66: Cassirer discerns in the recent development of the exactsciences:

1 A transition from the ‘species’ or ‘generic concept’ (Gattungsbegriff), to the concept ofFunction (Funktionsbegriff). The ‘generic concept’, according to the traditional logic (ofAristotle), is formed through abstraction; from the particulars of the individuals fallingunder it, those common features belonging to all are isolated. Whereas the concept offunction . . . stems from the new ‘logic of the mathematical concept of function’.

1 (Cassirer, Substance and Function, p. 21)

1 There is a strikingly wide variety of contexts in which the notion of ‘function’ operated in theearly years of this century. Frege used it to signify the sense of incomplete or ‘unsaturated’expressions, such as predicate expressions. When provided with an ‘argument’, i.e. the senseof a referring expression, a function generates the content of a thought, true or false. ThusFregean functions, like mathematical functions, have objects as values. Russell, in introduc-ing the idea of a ‘propositional function’, intended something similar. William James, though,used the notion of ‘function’ in a very different way; for him consciousness was a ‘function’,and we need, therefore, a functionalist theory of mind. For comments on the relation betweenfunctionalism in this sense, and structuralism, see S. Shapiro, Philosophy of Mathematics:Structure and Ontology (Oxford, Oxford University Press, 1997) pp. 106–8.

40 E. Cassirer, ‘Kant und die moderne Mathematik’, Kant-Studien 12 (1907): 7. The passage isquoted in T. A. Ryckman, ‘Conditio sine qua non? Zuordnung in the early epistemologiesof Cassirer and Schlick’, Synthese 88 (1991): 63, which draws attention to Russell’s commentin his Principles of Mathematics, second edition. (London, George Allen & Unwin, 1937),263–4:

1 In its most general form, functionality does not differ from relation . . . It is important. . . to observe that propositional functions . . . are more fundamental than other func-tions, or even than relations. For most purposes, it is convenient to identify the func-tion and the relation, i.e. if y = f(x) is equivalent to xRy, where R is a relation, it isconvenient to speak of R as the function . . . the reader, however, should rememberthat the idea of functionality is more fundamental than that of relation.

subject-predicate judgements, that the relation of a ‘thing’ to its propertieshas primacy, we must make relations themselves the focus of our thinking.What matters about a ‘thing’, whether it be mathematical or physical, is notwhat it is in itself but how it relates to other ‘things’ and how it functions –that is to say how it is co-ordinated – with respect to them. Helping himselfto hindsight, Cassirer intends to show that:

the concept of function constitutes the general schema and model according towhich the modern concept of nature has been molded in its progressive his-torical development41 . . . The logical development of natural science tendsmore and more to a recognition that the original, naïve representations ofmatter are super�uous; at most we grant them the value of pictorial represen-tations, and recognise the quantitative relations, that prevail between phenom-ena, as what is truly substantial in them.

‘Naïve’ representations of matter are discarded not so much because wejudge them to be wrong as that they are unnecessary. And it is clear that hesees this development as entirely in keeping with Kant’s understanding ofscience, for ‘the Critique of Pure Reason clearly taught that all we know ofmatter are mere relations’.42 Elsewhere he says:

we must choose between . . . two views of the world: either with empiricism wemust assume as existent only what can be pointed out as an individual in thereal presentation, or with idealism, af�rm the existence of structures, which con-stitute the intellectual conclusion of certain series of presentations, but whichcan never themselves be directly presented.

The empiricist, in Cassirer’s view, is the person who imagines ‘behind theworld of perceptions a new existence built up out of the materials of sen-sations’. Reality is, then, a logical construction out of sensations. This isclearly a reference to the positivism of Mach. But the idealist – perhapsbetter ‘critical idealist’ – ‘traces the universal intellectual schemata, in

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41 E. Cassirer, Substance and Function, and Einstein’s Theory of Relativity, translated by W. C.Swabey and M. C. Swabey (New York, Dover Publications, 1953) p. 21. Originally publishedas Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erken-ntniskritik (Berlin, Bruno Cassirer, 1910) and as Zur Einsteinschen Relativitätstheorie:Erkenntnis theoretische Betrachtungen (Berlin, Bruno Cassirer, 1921) Cf. D. Howard, ‘Rel-ativity, Eindeutigkeit, and monomorphism: Rudolf Carnap and the development of the cat-egoricity concept in formal semantics’, in R. N. Giere and A. W. Richardson (eds) Originsof Logical Positivism, Minnesota Studies in the Philosophy of Science, vol.16 (Minneapolis,Minnesota University Press, 1996) pp. 133–6.

42 Cassirer, op.cit. (note 41) p. 261. Cassirer is quoting from Kant’s Critique of Pure Reason,B.341:

1 All that we know in matter is merely relations (what we call the inner determinationsof it are inward only in a comparative sense), but among these relations some are self-subsistent and permanent, and through these we are given a determinate object.

which the relations and connections of perceptions can be perfectly rep-resented’. So, for an idealist, ‘the objects of physics: matter and force, atomand ether can no longer be misunderstood as so many new realities forinvestigation, and realities whose inner essence is to be penetrated’.Rather: ‘Atom and ether, mass and force are nothing but examples of suchschemata, and ful�ll their purpose so much the better, the less they containof direct perceptual content’. Science is not concerned, as the empiricistsupposes, with the ‘reality of things’; but the theoretical concepts it uses,though they have no direct intuitive content, ‘have a necessary function inthe shaping and construction of intuitive reality’. They are able to functionin this way because they are concerned not with the ‘perceptible proper-ties of the empirical objects, like their colour or taste’, but with ‘relationsof these empirical objects’.43

We need to appreciate that Cassirer’s Kant is a generalized Kant. He fullyaccepts that no philosopher can claim any longer that the principles of New-tonian mechanics or of Euclidean geometry are in some sense indispensableor otherwise privileged. But like his Marburg colleagues he wants to claim,as Kant had, that the exact sciences provide our paradigm of real objectiveknowledge. And he also wants to claim that something is indispensable forsuch knowledge, and he believes that the implication of recent develop-ments in the mathematical and physical sciences is that this indispensablesomething is structural and relational rather than substantial. It is, as he putsit, a ‘universal invariant theory of experience’ in which ‘the attempt is madeto discover those universal elements of form, that persist through all changein the particular material content of experience’. Using language that isclearly Kantian in tone, he continues: ‘The “categories” of space and time,of magnitude and the functional dependency of magnitudes, etc. are estab-lished as such elements of form, which cannot be lacking in any empiricaljudgment or system of judgments’.44

Not surprisingly, we find that in pursuing his aim Cassirer traces manyof the steps towards structural scientific realism. He uses, for example,Helmholtz’s theory of signs: ‘our sensations and presentations’, he says,‘are signs (Zeichen), not copies (Abbilden) of objects’. Signs, unlike copies,do not need to be in any way similar to what they signify or designate. Butif this is so, how can experience, and the sciences we construct with its help,inform us about objects? The answer to this crucial question is, as Cassirerexpresses it, that ‘what is retained in [the sign] is not the special characterof the signified thing, but the objective relations, in which it stands toothers like it’. He adds: ‘The manifold of sensations is correlated with themanifold of real objects in such a way, that each connection, which can be

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43 Cassirer, op. cit. (note 41) pp. 123, 165–6, 229.44 Cassirer, op. cit. (note 41) pp. 268–9. Cf. A. W. Richardson, ‘Logical idealism and Carnap’s

construction of the world’, Synthese 93 (1992): 67.

established in one group, indicates a connection in the other’.45 So, inaccordance with an abstract analytic treatment of physical theory, the con-cepts used in the exact sciences are conventional signs rather than naturalrepresentations of reality.46 These sciences, therefore, do not inform usabout real objects; but nevertheless even though the correlation or co-ordination (Zuordnung) between theory and reality is conventional ratherthan natural, the relational structure of reality will be represented intheory. Theories in the exact sciences, then, can inform us about real struc-tural relations though not about how these structural relations are real-ized. Thus Cassirer says:

we do not know, indeed, the real absolutely in its isolated, self-existent prop-erties, but we rather know the rules under which this real stands and in accord-ance with which it changes. What we discover clearly and as a fact without anyhypothetical addition is the law in the phenomenon. 47

For example, we do not know, or need to know, what chlorine is; it isenough if we know that, in appropriate circumstances, it reacts with othersubstances, such as sodium, in law-like ways. This knowledge is fully objec-tive. It is, as Cassirer explains, knowledge of real relations: ‘The objects ofphysics are thus, in their connection according to law, not so much ‘signsof something objective’ as rather objective signs, that satisfy certain con-ceptual conditions and demands’. ‘It follows’, he concludes, ‘that we neverknow things as they are in themselves, but only in their mutual relations’.48

How, though, is this claim to knowledge of real relations, or functionalcapacities, justified?49 For Cassirer, an implication of what had been takingplace in mathematics and the exact sciences in the nineteenth centurywas that Kant’s faculty of pure intuition must be rejected. Mathematicsand geometry, according to Dedekind and Hilbert, were not ‘about’ intu-ited mathematical and geometrical entities. Dedekind, for example,together with Cantor and Weierstrass, had sought to establish the rigour

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45 Cassirer, op. cit. (note 41) p. 304. Cf. T. A. Ryckman, ‘Conditio sine qua non? Zuordnungin the early epistemologies of Cassirer and Schlick’, Synthese 88 (1991): p. 69.

46 It was Heinrich Hertz who, in his Bildtheorie, applied Helmholtz’s Zeichentheorie to theor-etical science.

47 Cassirer, op. cit. (note 41) p. 304. Thus, if we wanted to picture the fact that Edinburgh isnorth of London we would use a symbol to represent Edinburgh and another symbol to rep-resent London, and these two symbols would have to be related to each other in a way whichserves to indicate the geographical relation of Edinburgh and London. We might, forexample, place the symbol representing Edinburgh on top of the symbol representingLondon.

48 Cassirer, op. cit. (note 41) p. 304.49 The function, or functional capacity, of an object, in Cassirer’s terminology, is identi�ed in

terms of its relation to other objects. Thus, chlorine has a functional capacity to combinewith sodium.

of mathematical analysis by dispensing with any appeal to intuition ofmathematical entities or to intuitive certainties in definitions of them.After suggesting that Dedekind appears to base the concept of purenumber on ‘the traditional logical doctrine of a plurality of things’, Cas-sirer points out that these ‘ “things” . . . are not assumed as independentexistences present anterior to any relation, but they gain their whole being. . . first in and with the relations which are predicated of them’. This isnot because there is something especially mysterious about these ‘things’which in itself makes intuition of them problematic; rather, if we followDedekind’s approach, they ‘are terms of relations and as such can neverbe “given” in isolation but only in ideal community with each other’. As aresult of the new sensitivity to rigor in mathematics, ‘the claim to grasp thesubstance of things in number has indeed been gradually withdrawn’.50

There is then no role for intuition in arithmetic, for according to thenew way of thinking ‘what is . . . expressed is just this: that there is asystem of ideal objects whose whole content is exhausted in their mutualrelations. The “essence” of the numbers is completely expressed in theirpositions’.51

This rejection of intuition is coupled with the claim that the new logic ofFrege and Russell, by placing a greater value on ‘relation-concepts’ thanon ‘thing-concepts’, requires that the a priori ground or condition of objec-tive knowledge in the exact sciences should stress the role of structure and

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50 Cassirer, op. cit. (note 41) pp. 36, 27. ‘More and more’, Cassirer says, ‘the tendency ofmodern mathematics is to subordinate the ‘given’ elements as such and allow them noinfluence on the general form of proof’ (Cassirer, op. cit. (note 41) p. 99). Cf. A. W.Richardson, ‘Logical idealism and Carnap’s construction of the world’, Synthese 93 (1992):64.

51 Cassirer, op. cit. (note 41) p. 39. The same point is made with regard to geometry:

1 For all these propositions [of non-Euclidean geometry] only express a system of rela-tions, while they make no final determination of the character of the individualmembers, which enter into these relations. The points, with which they are concerned,are not independent things, to which in and for themselves certain properties areascribed, but they are merely the assumed termini of the relation itself and gainthrough it all their character.

1 (Cassirer, op. cit. (note 41) p. 110)

1 See also Cassirer’s comment on Hilbert’s geometry as ‘pure theory of relations’:

1 The determination of the individuality of the elements is not the beginning but the endof the conceptual development; it is the logical goal, which we approach by the pro-gressive connection of universal relations. The procedure of mathematics here pointsto the analogous procedure of theoretical natural science, for which it contains the keyand the justi�cation.

1 (Cassirer, op. cit. (note 41) p. 94)

relations.52 It requires this because our experience results in knowledgethat is objective only in so far as it is grounded in this logic. So, just as ingeometry ‘the real object of geometrical interest is seen to be only the rela-tional connection between the elements as such, and not the individualproperties of those elements’, so also ‘the conceptual construction of exactphysics proves to be dominated by a corresponding logical procedure’. Thisis not to deny, of course, that ‘experiment is necessary to analyse an orig-inally undifferentiated perceptual whole into its particular constitutiveelements’, but nevertheless:

To mathematical theory belongs the determination of the form by which theseelements are combined into a unity of law. The system of ‘possible’ relationalsyntheses already developed in mathematics affords the fundamental schemafor the connections, which thought seeks in the material of the real. As to whichof the possible relational connections are actually realized in experience,experiment, in its result, gives its answer.53

Plainly, Cassirer’s structuralism is motivated by the conviction that theinsights of Kant’s transcendental logic, once divested of their dependenceon superseded natural philosophy, can legitimate the exact sciences as par-adigms of knowledge. They do this by virtue of the emphasis they place onthe role of logical form in ensuring the objectivity of knowledge in the exactsciences. A Kantian critical philosophy will thus provide a ‘logic of objec-tive knowledge’.54 This logic – the logic of Frege and Russell – in supplyingthe structure required for objectivity provides formal components of know-ledge. But for Cassirer, as for Kant, the structure supplied by logic is highlygeneralized. It is the structure of any possible mathematical physics yield-ing objective knowledge which is real, not the structure of some speci�ctheory in the exact sciences. For Cassirer, the reality revealed by the math-ematical sciences is a reality of abstract structures. Such an austere version

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52 See, for example, B. Russell, The Principles of Mathematics, second edition (London,George Allen & Unwin, 1937) p. 8:

1 Speaking generally, we ought to deal, in every branch of mathematics, with any class ofentities whose mutual relations are of a speci�ed type; thus the class, as well as the par-ticular term considered, becomes a variable, and the only true constants are the types ofrelations and what they involve.

1 Cf. Cassirer, op. cit. (note 41) p. 40:

1 Wherever a system of conditions is given that can be realized in different contents, therewe can hold to the form of the system itself as an invariant, undisturbed by the differenceof the contents, and develop its laws deductively.

53 Cassirer, op. cit. (note 41) pp. 251, 257. Cf. T. A. Ryckman, ‘Conditio sine qua non? Zuord-nung in the early epistemologies of Cassirer and Schlick’, Synthese 88 (1991): 61.

54 E. Cassirer, ‘Kant und die moderne Mathematik’, Kant-Studien 12 (1907): 44.

of structural realism did not, though, survive the scrutiny of those of Cas-sirer’s contemporaries who were to become founders of logical empiricism.This is not at all surprising given that there were others, whose views wereimportant for logical empiricism and who had been developing the idea ofstructural realism in a very different way.55

4

When Substance and Function was published in 1910, Moritz Schlick(1882–1936) was 29 years old and beginning to make his reputation as anindependently-minded philosopher. After having spent three years study-ing philosophy at the University of Zurich, he was appointed to his �rst aca-demic position at Rostock in 1910, and in the same year he published his�rst substantial philosophical papers in the theory of knowledge and phil-osophy of science. The call to succeed Mach and Boltzmann in the Chair ofthe History and Philosophy of the Inductive Sciences in the University ofVienna lay some years in the future. So did the Thursday evening meetingswhich he initiated shortly after he accepted that chair and which we nowassociate with the Vienna Circle, though some of those who were to becomemembers of that circle, such as Otto Neurath, the Hahns, and Philip Frankwere already meeting and discussing ideas they were to take up later.Carnap was an undergraduate studying philosophy and mathematics in Jenaunder Frege and Bruno Bauch, a prominent neo-Kantian. Wittgenstein wasstill in Manchester, studying engineering.56

In 1904 Schlick had completed a doctoral dissertation, dealing with there�ection of light in a non-homogeneous medium, under the supervision ofMax Planck in Berlin, and it was no doubt his background in the physicalsciences which led him, early in his thinking, to pay attention to their philo-sophical aspects and implications.57 It is entirely possible, too, that he willhave had his interest stimulated by Einstein who began his academicappointment at the University of Zurich in 1909, when Schlick was writinghis �rst substantial philosophical paper entitled ‘The nature of truth inmodern logic’. As its title indicates, this is an ambitious and comprehensivestudy of the nature and criteria of truth in both factual and conceptual

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55 For further discussion of Cassirer’s view and an account of their relevance in Carnap’s earlyphilosophy, see A. W. Richardson, ‘Logical idealism and Carnap’s construction of the world’,Synthese 93 (1992): 63–70, and A. W. Richardson, Carnap’s Construction of the World: TheAufbau and the Emergence of Logical Empiricism, (Cambridge, Cambridge UniversityPress, 1998), ch.5.

56 There is some information about Schlick’s life in E. T. Gadol (ed.) Rationality and Science:A Memorial Volume for Moritz Schlick in Celebration of the Centennial of His Birth (Viennaand New York, Springer–Verlag, 1982).

57 The title of Schlick’s dissertation was Ueber die Re�exion des Lichtes in einer inhomogenenSchicht, Berlin, 1904.

judgements. Like others writing about truth at this time, Schlick expressesdissatisfaction with the theories of truth then on offer. In particular, herejects what he describes as the ‘oldest and most natural account of thenature of truth . . . according to which it consists in a correspondence’. Thedif�culty is, he says, that the correspondence in question is that between‘thought’ and ‘things’, and these ‘things’ have usually been understood asKantian ‘things-in-themselves’ . On such an understanding however, it isquite clear that because of the ‘unknowability’ of these transcendent things-in-themselves the truth of any ‘thought’ would always be beyond our reach.And we are no better off, Schlick claims, if the ‘things’ with which ‘thought’must correspond are understood as accessible to experience. For that meansthat they will be ‘thoughts’ themselves, and then truth would become amatter of ‘correspondence of thought with itself’; in effect, the truth orotherwise of a ‘thought’ would be determined by its coherence with other‘thoughts’. At the heart of these familiar problems is, of course, the real-ization that the different versions of the correspondence theory of truth‘suffer chie�y from an inadequate elucidation of the concept of correspon-dence they employ’.58

What, then, does Schlick propose as a suitable de�nition of truth? Hisanswer is surprising and, at �rst sight, unpromising. He begins with the claimthat just as mental sensations and ideas are subjective signs designating realor conceptual things, such as snow or the number two, so the judgements orpropositions we consider should be thought of as subjective signs for theorder or form or structure of these things.59 ‘Judgments’, he says in hisGeneral Theory of Knowledge, ‘are signs for relations among objects’.60

Thus, the (true) judgement that snow is cold is a sign for the structure ofthe real fact that snow is cold, and the (true) judgement that 2 3 2 = 4 is asign for the structure of the conceptual fact that 2 3 2 = 4. Those sequencesof words we call declarative sentences are, of course, signs of judgements orpropositions. In accordance with Helmholtz’s Zeichentheorie, the conceptsoccurring in judgements are co-ordinated or correlated with the real objectsthey designate, but they do not correspond with them in the sense of beingimages or pictures of them. ‘We must’, Schlick says, ‘beware of supposing a

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58 M. Schlick, ‘The nature of truth in modern logic’, in H. L. Mulder and B. F. B. van de Velde-Schlick (eds) Moritz Schlick: Philosophical Papers, vol.1 (1909–1922), translated by P. Heath(Dordrecht, D. Reidel Publishing Company, 1979) pp. 60–1, 68. Originally published as ‘DasWesen der Wahrheit nach der modernen Logik’, in Vierteljahrsschrift für wissenschaftlichePhilosophie und Soziologie 34 (1910): 386–477.

59 Schlick, op. cit. (note 58), p. 91.60 M. Schlick, General Theory of Knowledge. Library of Exact Philosophy XI, translated by A.

E. Blumberg (Vienna, Springer-Verlag, 1974), p. 40. Cf. Schlick op. cit. (note 58), p. 91:

1 All judgements serve to designate for us the form of what is given in experience, in thesame sense as sensations and ideas designate for us the content of experience. . . . Justas ideas stand for the ‘what’, for things and properties, so judgments stand for facts.

judgement to be something more than a mere sign for the form of the ma-terial, and to be capable of describing this form adequately in somefashion’.61 And similarly, the relations between real things which are co-ordinated or correlated with the judgements which designate them do notcorrespond with relations between concepts in the sense that the latter areimages or pictures of them. ‘For’, Schlick explains, ‘temporal aspects alwaysenter into [relations between real things], and usually spatial aspects as well,whereas conceptual relations are non-temporal and non-spatial’. So, ‘in thejudgement “The chair is to the right of the table”, the concept “chair” is notplaced to the right of the concept “table” ’.62

The �rst dif�culty with this way of proceeding is that if judgements areno more than subjective private signs then they cannot be regarded as pos-sessing truth or falsity. They will only be true or false in so far as they canbe understood to relate to an ‘object’, i.e. can be understood as objective.How does thought, expressed in judgements, come to have this characterand to be, as we say, about reality? If the reality is metaphysical, as in thecase of Kant’s things-in-themselves, then because of the unbridgeable gulfbetween thought and such a reality objective judgements will be impossible.We need a conception of reality which will make objective judgement poss-ible. And as we have seen, according to neo-Kantianism, as expressed byCassirer for example, we obtain that conception by applying the a priorirules of pure logic and pure mathematics to the subjective data of experi-ence. It is, then, in the idealizations of mathematical natural sciences thatwe will �nd the objectivity we seek; the conception of reality those sciencesprovide – an idealized reality – is essentially logical and mathematical.These sciences will, of course, contain judgements which incorporate signsdesignating the content of experience, but the judgements themselves willbe signs of the structure of experience and will, if they are to be objective,be logical and mathematical signs of that structure. It is plain that the objec-tivity of judgements, and the reality that objective judgements designate, isa matter of their structure of form, rather than their content.

Schlick’s concern, though, is not so much with the objectivity of judge-ments as with their truth. If judgements are signs of states of affairs, and if,as Schlick assumes, such judgements are the primary bearers of truth andfalsehood, it seems that the difference between a true judgement and a falseone will be a difference between two kinds of sign. Some signs of states ofaffairs will be true judgements and some will be false judgements. But,Schlick says, the only kinds of sign that there can be are those that desig-nate ‘univocally’ and those that do not. For example, a road traf�c sign des-ignating a junction can, and should be, unambiguous, but it can be, andsometimes is, ambiguous. So the difference between true and false judge-ments must coincide with the difference between univocal and equivocal

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 97

61 Schlick, op. cit. (note 58) p. 93.62 Schlick, op. cit. (note 60) p. 61.

signs, and we are naturally drawn to the conclusion that a true judgementis one that designates univocally, and a false judgement is one that desig-nates equivocally: ‘a judgment that uniquely designates a set of facts is calledtrue’.63 But this way of de�ning truth does not seem right. Univocality in aproposition is, in many and perhaps all circumstances, a virtue but it is notthe virtue of truth. Othello’s belief that Desdemona loves Cassio is not falsebecause it lacks univocality; it is not an equivocal belief, and Iago’s beliefthat she loves no-one but Othello is not true because it is univocal. To quoteone recent commentator on Schlick’s account of the nature of truth andfalsity, ‘the existence of true but ambiguous propositions is suf�cient toshow how silly is this way of conceiving the nature of falsehood’.64

However, this objection misunderstands Schlick’s view. Truth is not amatter of linguistic ambiguity or equivocation – or lack of it. Judgementsare signs, but they are not linguistic signs. The univocality of a judgementis a matter of one-to-one co-ordination between the judgement and what itdesignates. We have, on the one hand judgements or propositions, and onthe other hand facts, and where there is a one-to-one co-ordination betweena judgement and a fact then the judgement is true. According to the famil-iar view, the truth of a judgement consists in its ‘correspondence’ or ‘agree-ment’ with the facts, but the central puzzle with this view is what is meantby ‘correspondence’ or ‘agreement’. Schlick’s claim is that there can benothing more to correspondence than co-ordination.65 So, when we say thata judgement is true because it corresponds with the facts, we must mean thatit is true because it co-ordinates, univocally, with the facts; and when we saythat a judgement is false because it fails to correspond with the facts, wemust mean that it is false because it fails to co-ordinate, univocally, with thefacts. So of course a linguistically ambiguous sentence can be true – and willbe true when one of the judgements it ambiguously expresses co-ordinatesunivocally with the facts. Failure of unique designation, or univocal co-ordi-nation entails that the co-ordination is equivocal. For example, Schlickexplains, a prediction is ‘a sign for an expected set of facts foreseen in

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63 Schlick, op. cit. (note 60) p. 60. Cf. Schlick, op. cit. (note 58) p. 94: ‘A judgment is true if itunivocally designates a speci�c state-of-affairs’.

64 D. Howard, ‘Relativity, Eindeutigkeit, and monomorphism: Rudolf Carnap and the develop-ment of the categoricity concept in formal semantics’, in R. N. Giere and A. W. Richardson(eds) Origins of Logical Positivism, Minnesota Studies in the Philosophy of Science, vol.16,(Minneapolis, Minnesota University Press, 1996) p. 126; cf. J. Alberto Coffa, The SemanticTradition From Kant to Carnap: To the Vienna Station, in Linda Wessells (ed.) (Cambridge,Cambridge University Press 1991) pp. 177–9.

65 Thus Schlick says:

1 the nature of truth does really rest on a correspondence; but by this we should wish tounderstand no more than the one-to-one coordination of judgments with facts . . . Inso saying, we think we have merely brought out what everyone has a more or less clearidea of, when they explain truth as correspondence of thought with its objects.

1 (Schlick, op. cit. (note 58) p. 99)

imagination’ and is ‘a sign also for the set of facts that actually appears’. Ifthese two sets of facts – anticipated and actual – are different and the signis therefore ambiguous between them, then we would quite naturally saythat the prediction is false.66 Clearly, for Schlick facts are what judgementsdesignate irrespective of whether the judgements are true or false. ‘In ajudgement’, he says, ‘we always think of a designation as having actuallybeen carried out, a co-ordination as having been consummated’.67 If thejudgement is true then the facts it designates coincide with the ‘existing’facts and the judgement is therefore univocal and designates uniquely. If thejudgement is false then the facts it designates differ from the ‘existing’ factsand the judgement is therefore equivocal and does not designate uniquely.

In the case of science, then, where judgements take the form of theories,we seek to provide information about the form or relational structure ofreality, rather than about its content. This distinction, between form andcontent, is one that Schlick is able to sustain with the aid of his well-knowncontrast between conceptual knowledge and intuitive acquaintance.68 Asagainst those who claimed that genuine knowledge is only possible where wehave direct acquaintance with, or ‘intuition’ of, the subject matter or contentof judgements, Schlick claims that it is a mistake to con�ate knowledge withacquaintance. There is no such thing as knowledge by acquaintance; instead,there is a distinction between knowledge on the one hand, and acquaintance

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66 Schlick, op. cit. (note 60) p. 62. He gives as another example the false judgment ‘A light rayconsists of a stream of rapidly moving particles’:

1 By examining all the facts taught us by physical research, we soon become aware thatthis judgement does not provide a unique designation of the facts. That is to say, we�nd that two different classes of facts are coordinated to the same judgements, thattherefore an ambiguity is present. On the one hand, we have the facts that actually doinvolve moving corpuscles, as in the case of cathode rays; on the other hand, we havea different set of facts, namely, those of the propagation of light, designated by the verysame symbols. Moreover, at the same time different signs are coordinated to two iden-tical series of facts, namely, those of the propagation of light and those of wave propa-gation. Uniqueness is forfeited and the proof that this is so is also the proof that thejudgment is false.

1 (Schlick, op. cit. (note 60) p. 62)

1 In an early unpublished essay Schlick repeats this example; see M. Schlick, ‘What isknowing?’, in H. L. Mulder and B. F. B. van de Velde-Schlick (eds) Moritz Schlick: Philo-sophical Papers, vol.1 (1909–1922), translated by P. Heath (Dordrecht, D. Reidel PublishingCompany, 1979), p. 137. There is a brief, and less clear, account of false judgement in Schlick,op. cit. (note 58) p. 97.

67 Schlick, op. cit. (note 60) p. 65. An implication of this is that ‘a judgement designates notmerely a relation, but the existence of a relation’. Cf. Schlick, op. cit. (note 66) p. 135: ‘Thatwhich is expressed in a judgment is always a fact. Or, if the judgement happens to be false,at least an alleged fact.’

68 Cf. M. Friedman, ‘Helmholtz’s “Zeichentheorie” and Schlick’s Allgemeine Erkenntnislehre:early logical empiricism and its nineteenth century background’, forthcoming, p. 6: ‘Schlick’scentral distinction is between conceptual knowledge and intuitive acquaintance’.

on the other. Knowledge, unlike intuition or acquaintance, must involve theapplication of concepts. In knowledge we always ‘put two objects into rela-tion with one another’, whereas in acquaintance ‘we confront just one object’.For, ‘so long as an object is not compared with anything, it is not incorpor-ated in some way into a conceptual system, just so long it is not known’. Ofcourse the content, or subject matter, of scienti�c theories is not accessibleto us in any direct sense; we are not ‘acquainted’ with it. But this is not initself a problem for even if we were acquainted with that content, we wouldstill lack knowledge. Instead we use the claims of theories as implicit de�-nitions of content: in theoretical physics, Schlick says:

it is a familiar fact that essentially different phenomena may nevertheless obeythe same formal laws. The same equation may represent quite different naturalphenomena depending on the physical meanings we assign to the quantities thatoccur in it . . . We conclude that a strictly deductive construction of a scienti�ctheory . . . has nothing to do with the intuitive picture we form of the primitiveconcepts.

In effect, scienti�c knowledge, like all knowledge, must be propositional; itmust therefore involve the application of concepts. And as conveyed in thetrue judgements which constitute theories in the exact sciences, proposi-tional knowledge takes the form of equations. In so far as we have know-ledge of what electricity, or gravitation is, or what atoms are, it will dependsolely on the relational claims of theories about electricity, or about atoms,as expressed in scienti�c laws. So, as Schlick puts it: ‘Maxwell’s equationsdisclose to us the “essence” of electricity, Einstein’s equations the essenceof gravitation. With their help, we are able in principle to answer all ques-tions that can be raised with regard to the objects of nature’.69

What, though, of this reality whose structure is described by true scien-ti�c theories? Is it, or rather is its structure, to be understood as an exter-nal and ‘mind-independent’ truth-maker for the judgements of scienti�ctheories? Though Schlick was, and remained a scienti�c realist, his answerto this question cannot be an unquali�ed yes. He is enough of a Kantian toappreciate that a ‘dogmatic’ realism which does not help us understand howour cognitive powers are able to grasp reality cannot be satisfactory. Somescienti�c principles must be imposed by us on reality, rather than by realityon us, if our experience and consequent scienti�c theorizing are to count asexperience of and theorizing about an objective reality. There are, that is tosay, principles which are constitutive of the objects of experience andtheory. We are able to cognitively grasp reality because some of the featuresor aspects of that reality, expressed in these constitutive principles, are ofour making. Kant had described these principles as having an a prioriguarantee, but Schlick is also enough of an empiricist to resist the thoughtthat there is anything in our cognitive grasp which is unrevisable in the light

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69 Schlick, op. cit. (note 60) pp. 82–3, 35–6, 242.

of experience. And in any case the speci�c principles which Kant hadthought a priori had been shown to be neither necessary nor even true.

One way of satisfying the pull of Kant and of empiricism is to adopt aversion of conventionalism as advocated by Poincaré, and there are someexplicit indications that this was the solution favoured by Schlick. Somecommentators have indeed attributed to him a view they call ‘realistic con-ventionalism’.70 According to this view some claims in scienti�c theories,namely the constitutive principles, will be accepted as true ‘by convention’.That is to say, those adopting the theory will regard the claims as true, notbecause the empirical evidence available determines their truth, or becauseour cognitive powers are such that they cannot but be accepted, but becausewe have chosen to regard them as true. Constitutive principles are not con-�rmed by evidence, as empirical laws are; indeed the adoption by conven-tion of these principles makes possible the con�rmation of empirical laws.71

But a Poincaréan conventionalism endorses Kant to the extent of admittingthat, as Poincaré acknowledged in The Value of Science, ‘a reality com-pletely independent of the mind which conceives it, sees or feels it, is animpossibility. A world as exterior as that, even if it existed, would be foreverinaccessible’.72 The world, or reality, must be ‘internal’ to the extent neces-sary to make it accessible, and it is our conventions which provide this ‘inter-nality’. Theories in the exact sciences provide us with information about thestructure of this world, and the constitutive principles incorporated intothose theories should be thought of as conventions selected by us, makingthe structure of reality dependent on us and, in that sense, internal.

My conclusion, then, is that structural realism is a theme that enables us tolink Schlick’s pre-positivism with Cassirer’s neo-Kantianism. But their routesto structural realism were very different: for Cassirer it is the consequence ofthe attempt to secure the objectivity of scienti�c knowledge; for Schlick it isthe outcome of his view about the nature of truth. The contrast between theirapproaches helps to illuminate, I suggest, the extent and nature of the Kan-tianism attributed to the founders of logical empiricism. Structural realism isa view readily associated with Kant, as Cassirer shows; but it is plain thatSchlick found himself drawn to such a view by the epistemology he devel-oped in conscious opposition to a positivistic kind of empiricism. But the con-vergence of Cassirer and Schlick on structural realism is not accidental. Bothmen seek to defend a realist’s understanding of the exact sciences; they do so

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70 D. Howard, ‘Realism and conventionalism in Einstein’s philosophy of science: the Ein-stein–Schlick correspondence’, Philosophia Naturalis 21 (1984): 619.

71 Reichenbach made this point by distinguishing between ‘axioms of coordination’ and‘axioms of connection’. The former, as constitutive principles, ‘have a completely differentstatus’ to the latter, as physical laws; see H. Reichenbach, The Theory of Relativity and APriori Knowledge, translated and edited by M. Reichenbach (Berkeley and Los Angeles,California University Press, 1965) p. 93. Originally published as Relativitätstheorie undErkenntnis Apriori (Berlin, Springer, 1920).

72 Poincaré, op. cit. (note 26) p. 209.

because in their view these sciences present us with a paradigm of objectiveknowledge. Neither is willing to endorse a naïve or ‘dogmatic’ realism whichclaims that by exercising our cognitive powers in accordance with the scien-ti�c method, we can achieve in scienti�c theories knowledge of a reality exist-ing quite independently of those cognitive powers.

It is important that the role of structural realism in the origins of logicalempiricism is not exaggerated. That role should be set alongside the otherthemes which have been identi�ed as playing a signi�cant part in the think-ing associated with those origins. In this paper I have been concerned toplace the discussion of structural realism by Cassirer and Schlick in theappropriate historical setting, but of course these ideas played an import-ant part in the development of Carnap’s thought and from a broader pointof view there can be little doubt that structural realism and logical empiri-cism are bound up with each other to a greater extent than I have indicated.This is not altogether surprising if we recall that Schlick and his fellowlogical empiricists were reacting to neo-Kantianism, especially its Marburgvariety. Schlick is severe in his attitude to the Marburg philosophers, andespecially so in his critique of Cassirer on relativity,73 but it would be sur-prising indeed if his opposition were to take the form of rejecting everythingthat he found in their work. As I have indicated, both Cassirer and Schlickwere working in an intellectual context where structural realism was a com-monly-held view about the exact sciences. Logical empiricism is, as its nameimplies, a variety of empiricism, but in so far as we can speak of its originsin the early thinking of Schlick we need to appreciate that it is a very highlyquali�ed empiricism, at least to the extent that it is capable of sustaining aversion of scienti�c realism. There is, though, a great deal more work thatneeds to be done to understand how, in the early years of this centuryepistemological positions were reconciled with the metaphysical commit-ments entailed by the physical sciences. In particular, we need to explorefurther the tensions within the conventionalism that was at the centre of thisissue. Given the powerful and lasting effect that logical empiricism has hadon philosophy of science in the twentieth century, the results of that explor-ation will make an important contribution to an understanding of ourcurrent preoccupations.74

Department of Philosophy, University of Durham

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73 See M. Schlick, ‘Critical or empiricist interpretation of modern physics’, in H. L. Mulder andB. F. B. van de Velde-Schlick (eds) Moritz Schlick: Philosophical Papers, vol.1 (1909–1922),translated by P. Heath (Dordrecht, D. Reidel Publishing Company, 1979) pp. 322–34. Orig-inally published in Kant-Studien 26 (1921): 96–111, as ‘Kritizistische oder empiristischeDeutung der neuen Physik’.

74 I am very grateful to a number of people for help in the preparation of this paper. I shouldespecially thank Ivor Grattan-Guinness, Thomas Uebel, Colin Howson and an anonymousreferee for good advice.

REFERENCES

Boyd, Richard. 1983. ‘The current status of scienti�c realism.’ Erkenntnis 19: pp.45–90.

Carnap, Rudolf. (1928) 1969. The Logical Structure of the World. Translated by R.George. (Originally published in 1928 as Der logische Aufbau der Welt. Berlin-Schlachtensee: Weltkreis-Verlag.) Berkeley, University of California Press.

Cassirer, Ernst. 1907. ‘Kant und die moderne Mathematik.’ Kant-Studien 12: pp. 1–49.Cassirer, Ernst. (1910/21) 1953. Substance and Function, and Einstein’s Theory of

Relativity. Translated by W. C. Swabey and M. C. Swabey. (Originally publishedin 1910 as Substanzbegriff und Funktionsbegriff: Untersuchungen über dieGrundfragen der Erkenntniskritik. Berlin, Bruno Cassirer, and in 1921 as ZurEinsteinschen Relativitätstheorie: Erkenntnis theoretische Betrachtungen. Berlin,Bruno Cassirer.) New York, Dover Publications.

Chihara, Charles S. 1990. Constructibility and Mathematical Existence. Oxford,Clarendon Press.

Dedekind, Richard. (1888) 1996. Was sind und was sollen die Zahlen? Translated byW. W. Beman, revised by William B. Ewald. Pp. 790–833 in From Kant toHilbert: A Source Book in the Foundations of Mathematics, vol. 2. Edited byWilliam B. Ewald. Oxford, Clarendon Press.

Demopoulos, William and Michael Friedman. 1985. ‘Critical notice: BertrandRussell’s The Analysis of Matter: its historical context and contemporary inter-est.’ Philosophy of Science. 52: pp. 621–39.

Demopoulos, William and Michael Friedman. 1989. ‘The concept of structure in TheAnalysis of Matter.’ Pp. 183–99 in Rereading Russell: Essays on BertrandRussell’s Metaphysics and Epistemology. Edited by C. Wade Savage and C. A.Anderson. Minnesota Studies in the Philosophy of Science, vol. 12. Minneapo-lis: Minnesota University Press.

Duhem, Pierre. (1906/14) 1954. The Aim and Structure of Physical Theory. Trans-lated by Philip P. Wiener. (Originally published in 1914 as the second editionof La Théorie Physique: Son Objet, Sa Structure. Paris: Marcel Rivière. The �rstedition was published in 1906.) Princeton, NJ., Princeton University Press.

Duhem, Pierre. 1911. Traité d’énergétique ou de thermodynamique générale. 2 vols.Paris, Gauthier-Villars.

Duhem, Pierre. 1996. Essays in the History and Philosophy of Science. Translatedand edited, with an introduction, by Roger Ariew and Peter Barker. Indi-anapolis and Cambridge, Hackett Publishing Co.

Dummett, Michael. 1991. Frege: Philosophy of Mathematics. London, Duckworth.Fine, Arthur. (1984) 1996. ‘Einstein’s realism.’ Pp. 86–111 in Arthur Fine, The Shaky

Game: Einstein, Realism, and Quantum Theory. Second Edition. (Originallypublished in Science and Reality: Recent Work in the Philosophy of Science,pp. 106–33. Edited by James T. Cushing, C. F. Delaney, and Gary M. Gutting.Notre Dame, University of Notre Dame Press.) Chicago, Chicago UniversityPress.

Frege, Gottlob. (1969) 1979. ‘Logic in mathematics.’ Pp. 203–50 in Gottlob Frege,Posthumous Writings. Translated by Peter Long and Roger White. (Originallypublished in 1969 as Nachgelassene Schriften und wissenschaftlicher Briefwech-sel, vol. 1. Edited by Hans Hermes, Friedrich Kambartel and FriedrichKaulbach. Hamburg, Meiner Verlag.) Oxford, Basil Blackwell.

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 103

Frege, Gottlob. (1976) 1980. Philosophical and Mathematical Correspondence. Trans-lated by Hans Kaal. (Originally published in 1976 as Nachgelassene Schriften undwissenschaftlicher Briefwechsel, vol. 2. Edited by Gottfried Gabriel, HansHermes, Friedrich Kambartel, Christian Thiel and Albert Veraart. Hamburg,Meiner Verlag.) Chicago, Chicago University Press.

Friedman, Michael. 1992. ‘Epistemology in the Aufbau.’ Synthese 93: pp. 15–57.Friedman, Michael. 1994. ‘Geometry, convention, and the relativized a priori:

Reichenbach, Schlick, and Carnap.’ Pp. 21–34 in Logic, Language, and the Struc-ture of Scienti�c Theories. Edited by Wesley C. Salmon and Gereon Wolters.Pittsburgh & Konstanz, Pittsburgh University Press and UniversitätsverlagKonstanz.

Friedman, Michael. Forthcoming. ‘Helmholtz’s “Zeichentheorie” and Schlick’s All-gemeine Erkenntnislehre: early logical empiricism and its nineteenth centurybackground.’

Gadol, E. T., ed. 1982. Rationality and Science: A memorial Volume for MoritzSchlick in Celebration of the Centennial of His Birth. Vienna and New York,Springer-Verlag.

Glymour, Clark. 1980. Theory and Evidence. Princeton, NJ., Princeton UniversityPress.

Gower, Barry. 1992. ‘Realism and empiricism in Schlick’s philosophy.’ Pp. 202–24 inScience and Subjectivity: The Vienna Circle and Twentieth Century Philosophy.Edited by David Bell and Wilhelm Vossenkuhl. Berlin, Akademie Verlag.

Gower, Barry. 1997. ‘Henri Poincaré and Bruno de Finetti: conventions and scien-ti�c reasoning.’ Studies in the History and Philosophy of Science 28: pp. 657–79.

Hallett, Michael. 1990. ‘Physicalism, reductionism and Hilbert.’ Pp. 183–257 in Phys-icalism in Mathematics. Edited by A. D. Irvine. Dordrecht, Kluwer AcademicPublishers.

Hellman, Geoffrey. 1990. ‘Modal-structural mathematics.’ Pp. 307–30 in Physicalismin Mathematics. Edited by A. D. Irvine. Dordrecht, Kluwer Academic Publish-ers.

Hempel, Carl. 1965. Aspects of Scienti�c Explanation. New York, Free Press.Hertz, Heinrich. (1892) 1962. Electric Waves: Being Researches on the Propagation

of Electric Action with Finite Velocity Through Space. Translated by D. E. Jones.New York, Dover Publications.

Hilbert, David. (1899) 1902. The Foundations of Geometry. Translated by E. J.Townsend. (Originally published in 1899 as ‘Grundlagen der Geometry’ inFestschrift zur Feier der Enthullung des Gauss-Weber Denkmals in Göttingen.Leipzig, Teubner.) Chicago, Open Court Publishing Co.

Hilbert, David. 1935. Gesammelte Abhandlungen . 3Bd. Berlin, Julius Springer Verlag.Howard, Don. 1984. ‘Realism and conventionalism in Einstein’s philosophy of

science: the Einstein–Schlick correspondence.’ Philosophia Naturalis 21: pp.616–29.

Howard, Don. 1993. ‘Was Einstein really a realist?’ Perspectives on Science 1: pp.204–51.

Howard, Don. 1996. ‘Relativity, Eindeutigkeit, and monomorphism: Rudolf Carnapand the development of the categoricity concept in formal semantics.’ Pp.115–64 in Origins of Logical Positivism. Edited by Ronald N. Giere and AlanW. Richardson. Minnesota Studies in the Philosophy of Science, vol. 16. Min-neapolis, Minnesota University Press.

104 BARRY GOWER

Jaki, Stanley. 1984. Uneasy Genius: The Life and Work of Pierre Duhem. The Hague,Martinus Nijhoff.

Kant, Immanuel. (1781/1787) 1965. Critique of Pure Reason. Translated by NormanKemp Smith. (Originally published in 1787 as the second edition of Critik derreinen Vernunft. Riga, Hartknoch. The �rst edition was published in 1781.)London, Macmillan.

Kant, Immanuel. (1783) 1953. Prolegomena To Any Future Metaphysics. Translatedby P. G. Lucas. Manchester, Manchester University Press.

Kitcher, Philip. 1993. The Advancement of Science: Science Without Legend, Objec-tivity Without Illusions. Oxford, Oxford University Press.

Laudan, Larry. 1981. ‘A confutation of convergent realism.’ Philosophy of Science48: pp. 19–48.

Leplin, Jarrett, ed. 1984. Scienti�c Realism. Berkeley, California University Press.Maxwell, Grover. 1970a. ‘Structural realism and the meaning of theoretical terms.’

Pp. 181–92 in Analyses of Theories and Methods of Physics and Psychology.Edited by Martin Radner and S. Winokur. Minnesota Studies in the Philosophyof Science, vol. 4. Minneapolis, Minnesota University Press.

Maxwell, Grover. 1970b. ‘Theories, perception and structural realism.’ Pp. 3–34 inThe Nature and Function of Scienti�c Theories. Edited by R. Colodny. Pitts-burgh, Pittsburgh University Press.

Natorp, Paul. 1910. Die logischen Grundlagen der exacten Wissenschaften. Wis-senschaft und Hypothese, vol. 12. Leipzig, B. G. Teubner.

Parsons, Charles. (1990) 1996. ‘A structuralist view of mathematical objects.’Pp. 272–309 in The Philosophy of Mathematics. Edited by W. D. Hart. (Orig-inally published in 1990 in Synthese 83: pp. 303–46.) Oxford, Oxford UniversityPress.

Poincaré, Henri. (1902) 1952. Science and Hypothesis. Translated by W. J. Green-street. (Originally published in 1902 as La science et l’hypothèse. Paris: Flam-marion.) New York, Dover Publications.

Poincaré, Henri. (1905) 1913. The Value of Science. Translated by George B.Halsted. Pp. 201–355 in Henri Poincaré, The Foundations of Science: Scienceand Hypothesis, The Value of Science, and Science and Method. (Originally pub-lished in 1905 as La valeur de la science. Paris, Flammarion.) Lancaster. Pa., TheScience Press.

Poincaré, Henri. (1908) 1914. Science and Method. Translated by Francis Maitland.(Originally published in 1908 as Science et méthode. Paris, Flammarion.)London, Thomas Nelson and Sons.

Psillos, Stathis. 1995. ‘Is structural realism the best of both worlds?’ Dialectica 49:pp. 15–46.

Reichenbach, Hans. (1920) 1965. The Theory of Relativity and A Priori Knowledge.Translated and edited by Maria Reichenbach. (Originally published in 1920 asRelativitätstheorie und Erkenntnis Apriori. Berlin: Springer.) Berkeley and LosAngeles, California University Press.

Resnik, Michael D. 1981. ‘Mathematics as a science of patterns: ontology and refer-ence.’ Noûs 15: pp. 529–50.

Resnik, Michael D. 1997. Mathematics as a Science of Patterns. Oxford, ClarendonPress.

Richardson, Alan W. 1992. ‘Logical idealism and Carnap’s construction of theworld.’ Synthese 93: pp. 59–92.

CASSIRER, SCHLICK AND ‘STRUCTURAL’ REALISM 105

Russell, Bertrand. (1903) 1937. The Principles of Mathematics. Second edition. (Firstedition published in 1903.) London, George Allen & Unwin.

Russell, Bertrand. 1919. Introduction to Mathematical Philosophy. London, GeorgeAllen & Unwin.

Russell, Bertrand. (1927) 1992. Analysis of Matter. London, Routledge.Ryckman, Thomas A. 1991. ‘Conditio sine qua non? Zuordnung in the early episte-

mologies of Cassirer and Schlick.’ Synthese 88: 57–95.Sauer, Werner. 1989. ‘On the Kantian background of neopositivism.’ Topoi 8: pp.

111–19.Schlick, Moritz. (1910) 1979. ‘The nature of truth in modern logic.’ Pp. 41–103 in

Moritz Schlick: Philosophical Papers, vol. 1 (1909–1922). Edited by Henk L.Mulder and Barbara F. B. van de Velde-Schlick. Translated by Peter Heath.(Originally published in 1910 in Vierteljahrsschrift für wissenschaftliche Philoso-phie und Soziologie 34: pp. 386–477, as ‘Das Wesen der Wahrheit nach der mod-ernen Logik.’) Dordrecht, D. Reidel Publishing Company.

Schlick, Moritz. (1910/11) 1979. ‘What is knowing?’ Pp. 119–40 in Moritz Schlick:Philosophical Papers, vol. 1 (1909–1922). Edited by Henk L. Mulder andBarbara F. B. van de Velde-Schlick. Translated by Peter Heath. Dordrecht, D.Reidel Publishing Company.

Schlick, Moritz. (1921) 1979. ‘Critical or empiricist interpretation of modernphysics.’ Pp. 322–34 in Moritz Schlick: Philosophical Papers, vol. 1 (1909–1922).Edited by Henk L. Mulder and Barbara F. B. van de Velde-Schlick. Translatedby Peter Heath. (Originally published in 1921 in Kant-Studien 26: 96–111, as‘Kritizistische oder empiristische Deutung der neuen Physik.’) Dordrecht, D.Reidel Publishing Company.

Schlick, Moritz. (1918/25) 1974. General Theory of Knowledge. Library of ExactPhilosophy XI. Translated by Alan E. Blumberg. (Originally published in 1925as the second edition of Allgemeine Erkenntnislehre. Berlin, Springer. The �rstedition was published in 1918.) Vienna, Springer-Verlag.

Shapiro, Stewart. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford,Oxford University Press.

Suppe, Frederick. 1989. The Semantic Conception of Theories and Scienti�c Realism.Urbana, Illinois University Press.

Tait, William W. 1997. ‘Frege versus Cantor and Dedekind: on the concept ofnumber.’ Pp. 213–48 in Early Analytic Philosophy: Frege, Russell, Wittgenstein:Essays in Honor of Leonard Linsky. Edited by William W. Tait. Chicago andLa Salle, Illinois, Open Court.

van Fraassen, Bas. 1980. The Scienti�c Image. Oxford, Oxford University Press.Worrall, John. (1989) 1996. ‘Structural realism: the best of both worlds?’ Pp. 139–65

in The Philosophy of Science. Edited by David Papineau. (Originally publishedin 1989 in Dialectica 43: 99–124.) Oxford, Oxford University Press.

Worrall, John. ed. 1994a. The Ontology of Science. Aldershot, Dartmouth.Worrall, John. 1994b. ‘How to remain (reasonably) optimistic: scienti�c realism and

the “luminiferous ether”.’ Pp. 334–42 in PSA 1994, vol. 1. Edited by David Hull,Mickey Forbes and Richard M. Burian. East Lancing, Michigan, Philosophy ofScience Association.

Zahar, Elie. 1996. ‘Poincaré’s structural realism and his logic of discovery.’ Pp. 45–68in Henri Poincaré: Science and Philosophy, International Congress Nancy,France, 1994. Edited by Jean Louis Greffe, Gerhard Heinzmann and KunoLorenz. Paris and Berlin, A. Blanchard and Akademie Verlag.

106 BARRY GOWER