Demopoulos - 1994 - Frege, Hilbert, And the Conceptual Structure of Model Theory

7/28/2019 Demopoulos - 1994 - Frege, Hilbert, And the Conceptual Structure of Model Theory

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HISTORY AND PHILOSOPHY OF LOGIC, IS (1994), 211-225

Frege, Hilbert, and the Conceptual Structure of Model Theory WILLIAM DEMOPOULOS

Philosophy Department, The University ofWestern Ontario, London, Ontario N6A 3K7, CanadaReceived 12 July 1992 Revised 12 April 1993

This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert'sGrund/agi!n tk r Gi!ometrie to two: (i) Frege's reliance on what, following Wilfrid Hodges, I call aFrege-Peano language, and (ii) Frege's view that the sense of an expression wholly determines itsreference. I argue that these two preronceptions prevented Frege from achieving the conceptualstructure of model theory, whereas Hilbert, at least in his practice, was quite close to themodel-theoretic point of view. Moreover, the issues that divided Frege and Hilbert did not revolvearound woether one or the other allowed metalogical notions. F rege, e.g., succeeded in formulatingthe notion of logical consequence, at least to the extent that Bolzano did; the point is rather thateven though Frege had certain semantic concepts, he did not articulate them model-theoretically,whereas, in some limited sense, Hilbert did.I. Introduction

In his definitive essay, 'The main trends in the foundations of geometry inthe 19th century', Hans Freudenthal summarizes the 'forerunners and competitors' of Hilbert's Grundiagen der Geometrie as follows:

. . . The father of rigor is Pasch. The idea of the logical status of geometryoccurred at the same time to some Italians. Implicit definition was analyzedmuch earlier by Gergonne. The proof of independence by counter-examplewas practiced by the inventors of non-Euclidean geometry, and more consciously by Peano and Padoa. The segment calculus was prefigured in vonStaudt's 'throw calculus.' The purport of the 'Schliessungssatze' had beengrasped by H. Wiener (1893). Even the title Grund/agen der Geometrie is farfrom original. Before Hilbert, a name like this indicated research likeRiemann's and Helmholtz's. Lie's papers of 1890 appeared under this title,and so did Killing's book in 1893, 1897.1Nevertheless, Freudenthal concludes that

. . . in spite of aU these historical facts we are [rightly] accustomed to identifythe tum of mathematics to axiomatics with Hilbert's Grund/agen: Thisthoroughly and profoundly elaborated piece of axiomatic workmanship wasinfinitely more persuasive than programmatic and philosophical speculationson space and axioms could ever be, (p, 619)If the mathematical basis for our assessment of the place of the Grund/agen inmodem axiomatics is clear, its methodological significance is only slightly less

Freudenthal, 1962. p. 619. All parenthetical page references to Freudenthal arc to this paper.OI..... ~ $10.00 C 19114 To)'lor A r ...... l.Id

212 William Demopoulosstraightforward, Again it is difficult to improve on Freudenthal's remark thatwith Hilbert

. . . the bond with reality is cut. Geometry has become pure mathematics. Thequestion whether and how to apply it to reality is the same in geometry as it isin other branches of mathematics. Axioms are not evident truths. They arenot truths at all in the usual sense. . . . Hilbert's clean cut between mathematics and realistic science became the paradigm of a new,methodology. (p. 6180rIn this paper I am concerned with a methodological aspect of the Grund-

lagen that is closely related to 'Hilbert's clean cut between mathematics andrealistic science', namely Hilbert's deployment of the category of expression thatwe today call non-logical constants. This use is indicated at the very beginning ofthe Grundlagen when Hilbert writes, 'We imagine three kinds of things '"called points . . . called lines . . . called planes . . . We imagine points, lines andplanes in some relations . . . called lyinf on, between, parallel, congruent. 2Developing a point of Wilfrid Hodges, I hope to show how, by treatingHilbert's presentation of the standard geometric vocabulary as an anticipation ofthe idea of a non-logical constant, it is possible to illuminate the central issue inhis correspondence with Frege, namely, their controversy over Hilbert'sapproach to proofs of independence.4 In arguing this thesis, I hope to make itplausible that virtually every point of controversy raised in the correspondence istraceable to their different perceptions of the nature and significance of thenotion of a non-logical constant. Hilbert may have misconstrued non-logicalconstants in some respects-here his belief that our use of such expressionscommits us to viewing axioms as a particular sort of definition comes especially2 It is signalled rather more dramatically in the correspondence with Frege when Hilbert writes:

. .. you say that my concepts, e.g. 'poi nt', 'betwe en', are not unequivocally fixed. 'Between' isunderstood differently on p. 20 and a point is there a pair of points. But it is surely obviousthat every theory is only a scaffolding (schema) of concepts together with their necessaryconnections, and that th e basic elements can be thou&ht of in any way one likes. E.g., insteadof points, think of a system of love, law, chimney-sweep .. . which satisfies aU the axioms;then Pythagoras' theorem also applies to these things. Any theory can alwllys be applied toinfinitely many systems of basic elements.Hilbert to Frege, 29.12.99, draft or excerpt by Hilbert. In Frege 1980, p. 42. Hilbert's pagereference corresponds to p. 28f, Section 9 of !be tenth edition of Grund/agi!n der Gwmnrii!.revised and enlarged by Paul Bemays and translated as Hilbert 1971.3 Hodges 1985/86. All reference& to Hodges are to this paper .4 A passage from Hilbert's second letter to Frege (29.12.1899) contains a summary of themathematical accomplishments of the Grun4/agi!fI which emphasizes the importance Hilbertattached to his trCltment of questions of independence:

[The work makes] it possible to understand those propositions that (arc the) most important resultsof geometrical inquiries: that the parallel axiom is not a consequence of the other axioms. andsimilarly Archimed es' axiom . . (ItJanswers the question whether it is possible to prove thatin two identical rectangles with an identical base line the sides must also be identical, orwhether as in Euclid this proposition is a new postulate. (It makes) it possible to understandand answer such questions as why the sum of the angles in a triangle is equal to two rightangles and how this fact is connected with the parallel axiom.The passage is excerpted by Frege, ct. Frege 1980, p. 38f. By 'identical rectangles' Hilbert heremeans rectangles of equal area; the relevant chapter for this result in the English translation isCb. IV, 'Theory of Plane Area'.
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Frege, Hilbert, and the ConceptUllI Structure of Model Theory 215distinction between non-logical constants and variables, is to show that thearguments go through when non-logical constants possessing a preanalyticmeaning ('line,' and 'between" for example) are replaced by 'letters' like 'RI7"i.e. by non-logical constants whose meaning is minimal in the sense justexplained. This of course is compatible with non-logical constants having morethan minimal meaning.On the present view, non-logical constants thus behave rather like theindexicals of a natural language, with the space of structures for the language ofthe theory playing a role analogous to the role played by space-time in fIXing thereference of inde:llical expressions like 'yesterday' and 'today'. Just as thereference of such an indexical is determined relative to a spatio-temporalcontext, the refereoce of a non-logical constant is determined once we are givena structure for the language. The situation is different with variables. Variablesare without meaning. In the course of evaluating the truth of a sentence in astructure, we may stipulate a reference for the variables relative to an arbitrarysequence of elements of the domain of the structure. But we do not regard thesequence as an essential feature of the structure. Not so the distinguishedelements, relations and operations which interpret the non-logical constantsthey are an essential feature of the structure.

It is interesting to compare our remarks on non-logical constants with Frege'sreaction to a paper of E. V, Huntington, one of the early exponents of the'modem' axiomatic method. 10 Around the time of his correspondence withHilbert, Frege' wrote to Huntington about his (i.e. Huntington's) use ofnon-logical constants, although Frege did not, of course, put the matter in theseterms. In the coone of the letter, Frege quotes a passage from Huntington'spaper 'A complete set of postulates for the theory of absolute continuousmagnitude': 'I f the first of two given elements is denoted by a and the second byb, then the object which they determine is denoted by a 0 b .'11 He thencomments:

Here the definite article in 'the object' seems to me incorrect. For what does'203' mean? We can assume different rules of combination. If we take thesum then 5 is determined; if we take the product, 6 is determined; and so wecan assume countless further rules, each of which determines a differentnumber; but we cannot tell from '20 3' which rule is assumed. If the signscontained some indication of the rule of combination, the matter might beb idifferent; but as it is 'a 0 does not mean anything. It seems to me that weare here preseuted with a cross between two different kinds of signs, namelythose that designate (or mean) and those that merely indicate . . . . [T]he sign

Indeed dun . : the period of elaboration of any deductive theory we cboose the ideas to be"presented by die undefined symbols and the facts 10 be stated by the unproved propositions:but, When we bl:Bin 10 formulate the theory, we can imagine that the undefined symbols are_ple te ly deWJid of meaning and that the unproved propositions instead of stating facts, thatis, relations between ideas represented by undefined symbols, are simply conditions imposedupon the undefillcd symbols.

(Padoa's emp/lases bv e been omitted.)IQ See Scanlan 1991_11 Huntington 1902t1. p. 266, quoted in Frege 1980, p. 58. The editon of the correspondence alsocite lite papen ~ o n 1901/19{)2 and Huntington 1902b.

216 William Demopoulos'0' seems to me to be a hybrid, for it is not a letter [i.e. a variable), and theway it is introduced makes us take it for a designating one. But we see that itdoes not designate anything, but is used more like an indicative sign [i.e. avariable). Its hybrid nature gives rise to unclarities about the sense of thepropositions in which it occurs. 12

Hodges has given an accessible explanation as to why Frege found non-logicalconstants so problematic: a language like that of the Begriffsschrift, or Peano'sFormulaire-a Frege-Peano language-has logical constants, variables, and arather restricted set of natural language expressions which occur with theirordinary meanings. Such a language elucidates the logical resources implicit innatural language by having a rigidly defined syntax, and thus constitutes alogically refined or 'regimented' fragment of the natural language. But it is not,and was not intended to be, a formal language of the sort with which we aretoday familiar. One of the features of natural language that simply has noanalogue in a Frege-Peano language is the presence of indexicals. Non-logicalconstants, being like indexical expressions in the way they function, would forthis reason alone make it difficult to express Huntington's axiomatization withina Frege-Peano language, and Hodges's account therefore rather naturallysuggests itself: Frege rejects the notion of a non-logical constant because hisbasic tool for the analysis of concepts- his 'concept-script' - ignores the categoryof expression to which they are most similar. Rejecting non-logical constants,Frege must reject Hilbert-style axiomatizations such as Huntington's, since inthis approach logical constants, variables, and non-logical constants exhaust thevocabulary of the language of the mathematical theory being axiomatized. I nowwant to elaborate this observation into an account of Frege's controversy withHilbert over the foundations of geometry.

3. Frege on independence results in geometryIn a letter to Hilbert dated 6.1.1900 Frege raises what he thinks is a generaldifficulty for Hilbert's approach to proofs of independence, although he addresses only the matter of the parallel postulate:.. there is a logical danger in your speaking of, for example, 'the parallelaxiom,' as if it was the same thing in every special geometry. Only thewording is the same; the thought content is different in every differentgeometry. It would not be correct to call the special case of Pythagoras'theorem [that deals with isosceles right triangles] the theorem of Pythagoras;for after proving that special case, one still has not proved Pythagoras'

theorem. Now given that the axioms in special geometries are all special cases12 Frege 1980, pp. 58t. The letter to Huntington is undated. Frege's editors conjecture that it waswritten in 1902. It is worth remarking that in his monograph, The funda_mal propositions ofmodern algebra, Huntington refers to '. ' as a variable (the term he uses is 'variable symbol') andcharacterizes his 'postulates' as propositional functions. with a refererwe to Russell's Principlesof I"I1Qlhemolics. (The monograph appeared in J. W. A. Young, ed . Monographs on topics ofmodern mathematics, second edition. Longmans, Green and Co. (New York) 1915; d. p. 172.The first edition, whiCh is not available to me, appeared in 1911.} However. Huntington doesnOI use this lenninology in connection with '0' in any of the papers cited in the correspondence.The editors' parenthetical addition to fn. 4, p. 58 is misleading rather than clarifying, insofar asit suggests that Huntington's later terminology was employed in the papen under discussion.
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217Frege, Hilbert, and the Conceptual Structure of Model Theoryof general axioms, one can conclude from lack of contradiction in a specialgeometry to lack of contradiction in the general case, but not to lack ofcontradiction in another special case.13

Thus, according to Frege, the proposition Hilbert shows independent of theother axioms concerns the 'general' concept of a line. But in the case ofEuclidean geometry, we are concerned with the 'special' case of Euclidean lines.From the fact that two propositions dealing with the general concept areindependent, we cannot infer that propositions which concern the Euclideancase are independent. Since every independence claim is equivalent to anassertion of consistency, this falls under an abstract discussion presented earlierin the letter, namely Frege's remark that

(i]f'a general proposition contains a contradiction, then so does any particularproposition that is contained in it. Thus if the latter is free from contradictionwe can conclude that the general proposition is free from contradiction, butnot conversely.14

This remark is clearly incorrect if we understand Frege to be comparinguniversal sentences with their instances. S However, the discussion of Pythagoras's theorem, which directly follows the remark, suggests that Frege's talk ofparticular propositions contained in a general proposition is not concerned withinstances but with propositions which make assertions about concepts of widerand narrower scope; the former propositions are called 'general', the latter,'particular'. The examples of right triangle and isosceles right triangle, line andEuclidean line, support this interpretation. Reading the passage in this way, wecan put Frege's point as follows: Hilbert has shown that it is consistent to assertthe axioms of incidence, order and congruence, together with the denial of theparallel postulate, of the general concept of a line; but from this it does notfollow that this may consistently be asserted of Euclidean lines, since were weto spell out the meaning of 'Euclidean line', this would require the parallelpostulate.Now it really doesn't matter whether or not we agree with F ~ e g e on the issueof whether Euclidean line presupposes the parallel postulate, since it is clearthat he has missed a crucial point. Anachronisms aside, we can put the matterthis way: On the modern, 'Hilbertian' view, the parallel postulate is true insome structures and false in others, and to see this, we simply don't need thenotion of a Euclidean line. But for Frege this simple answer is quite problematic.If sense determines reference, the general proposition can be true where theparticular proposition is false only if the sense of the general proposition (theproposition which concerns the general notion of a line) is different from thesenses of those propositions that refer to Euclidean lines. But then we areconstantly equivocating and never succeed in proving what was promised,namely, a result about the Euclidean axiom of parallels, which concernsEuclidean lines. But surely this just shows that we must give up anything like a13 Frege }980. p. 48. 14 Frege 1980, p. 47. IS Since 'Frege [then] appears to be saying that if (x)Fx is inoclDsistent. so is Fa for any a. [But)this is obviously not sol. Ljet"Fx" be "Ox &. ..,(,)0,,' ' ' Coffa 1986. p. 63. fn. 38.

218 William Demopoulosstrict adherence to Frege's view of sense and reference, if we are to understandHilbert's approach to axiomatics. Or to put the point slightly differently: inorder that Hilbert's axioms should be interpretable over Euclidean structures, itis not necessary that 'line', for example, should have the sense of 'Euclideanline'. The sense of 'line' is but one component in the determination of itsreference; the other component-the particular structure under considerationand the way in which it contributes to the reference of the primitives-is simplyignored by Frege's theory.This interpretation of Frege's remarks to Hilbert is reinforced by Frege's owndescription of the correspondence, conveyed several months later in a letter toHeinrich Liebmann. Frege writes:

I have reasons for believing that the mutual independence of the axioms ofEuclidean geometry cannot be proved. Hilbert tries to do it by widening thearea so that Euclidean geometry appears as a special case; and in this widerarea he can show lack of contradiction by examples; but only in this widerarea; for from lack of contradiction in a more comprehensive area we cannotinfer lack of contradiction in a narrower area; for contradictions might enterjust because of the restriction. The converse inference is of course permissible.Hilbert was apparently deceived by the wording. If an axiom is worded in thesame way, it is very easy to believe that it is the same axiom. But it dependson the sense; and this is different depending on whether the words 'point','line', etc. are understood in the sense of Euclidean geometry or in a widersense. 16Frege's difficulty with Hilbert's methodology thus stems from two sources: thecharacter of his concept-script and his conception of how the sense of adesignating expression determines its reference. Earlier we noted that theconcept-script, being a Frege-Peano language, has, aside from its logicalresources, only variables and non-indexical expressions; the latter have whateversenses are given them by the natural language to which they belong. The notionof sense appropriate for the expressions of a Frege-Peano language is one thatdetermines the reference of expressions independently of a structured context,whether this consists of locations in space-time, or of structures for a formalizedfragment of the language. While questions concerning the theory of truth for thesentences of such a language can be posed quite naturally, this is not the casefor the notion of truth in a structure. Even to pose the question of the correctdefinition of this concept requires the prior isolation of non-logical constantsexpressions which are not variables, but whose reference nevertheless varieswith changes in context. Such expressions differ fundamentally from Frege'sdesignating expressions, since for Frege the sense of a designating expressiondetermines its reference 'absolutely', i.e. independently of such contextualconsiderations as are relevant in the case of non-logical constants.I' In order to16 29.12.1900; Frege 1980. p. 91.17 Frege developed views on the sense of indexical expressions in his laiC essay 'Thoughts'; ct.Frege 1984, pp. 3S1-72. It has often been noted that these views are rather unintuitive if thesense of such an expression is identified with its linguistic meaning. As far as I can judge, theonly respect in which Frege is committed to some notion of the 'absoluteness' or 'fixity' ofreference is the one given in the text; By ignoring a central feature of the linguistic meaning ofindexicals and indexical-like expressions. Frege docs not sec the utility of a notion of reference


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221Frege, Hilbert, and the Conceptual Structure of Model Theoryis the extent to which Frege's discussion of der.:ndence is reminiscent ofBolzano's account of 'deducibility' [Ableitbarkeitj,2 and therefore suggestive ofTarski's 1936 formulation of the concept of logical consequence?'In a review of the first English translation of this series of papers, HowardJackson28 presented a rather natural reconstruction of a sufficient condition forindependence; Jackson's reconstruction is based on the notion of a pseudoproposition, which Frege introduced in the second Number of the second seriesof essays.29 Taking Hilbert's axioms as pseudo-propositions (expressed by openformulas containing individual, concept, and relation variables in place of thegeometric primitives) pseudo-axiom G is said to be independent of the (finitelymany) pseudo-axioms Q if the universal closure of the conditional, consisting ofG as consequent and the conjunction of the pseudo-axioms in g as antecedent,is false. Jackson observed that if Frege thought the notion of independencecould not be applied to Hilbert's work via the apparatus of pseudo-propositions,he was simply mistaken. On this Jackson is certainly right. But by focussing onan account in terms of pseudo-propositions, Jackson's discussion might suggestthat Frege did not significantly extend Russell's notion of formal implication. Ina similar vein, John Corcoran and Susan WoodJO remark that, given the contextof his debate with Hilbert, Frege's failure to formulate the notion of logicalconsequence is a 'curious nonevent in the history of logic'.Now in fact, in his discussion of dependence, Frege comes very close toformulating a notion of logical consequence. Frege is here concerned with theindependence of one 'real axiom' from another, where a real axiom is expressedby a sentence whose sense is wholly determinate and whose reference is fixed'absolutely'. Such an axiom is also true. Frege's idea31 may be explained asfollows.Let g U {G} be a set of propositions, expressed in a Frege-Peano language.We may think of the pair, (g , G) , as a 'premise-conclusion argument', with thepropositions in Q the premises, and G the conclusion, of the argument. (Inaccordance with our current practice, Frege suppresses explicit mention oflogical laws.) The only difference between this contemporary usage and Frege'sis that Frege appears to be interested only in the case where g is a set of truepremises. If g consists of the axioms of some special science, then the premisesmust be true, since the notion of a false axiom is an oxymoron for Frege.

'1fJ BoIzano 1837, section ISS, paragraph 2. (See also the Note to this section.) Frege is perhapscloser to BoIzano's notion of the ground-oonsequence relation (AbfolgeJ, which, for Balzano,holds between /I'Ulhs, rather than propositions in general.27 Tarsld 1956. As we noted earlier, the concepts of non-logical constant, structure, and truth in astructure make their first fully explicit appearance only in Tarski's post-war work. The distancebetween Tarski's paper (first published in Polish and German in 1936) and the model-theoreticanalysis of logical consequence is discussed in Etchemendy 199(). Etchemendy's principalhistorical point. centering on the divergence of Tarski's early notion of a model from our currentunderstanding, and the absence, in Tarski's 1936 paper, of the notion of a variable domain, wasnoted in Corcoran 1972p3. For related observations on Bolzano, see Section IX of Berg'sintroduction to his edition of BerMrd Bolzano: theory of science; 1973, pp. 20-22.28 Jackson, 1981.29 Modulo attention to use and mention, Frege's notion of a pseudo-proposition is evidently closelyrelated to Russell's notion of a propositional function, and may perhaps have been inspired by

it. 30 Corcoran and Wood 1972P3 31 See Frege 1984, pp. 335-9.

222 William DemopoulosHowever Frege does not require that the propositions in g must be known tobe true.For each type of non-logical expression E, i.e. for each concept or relationexpression, and for each proper name, we let E' be another meaningfulexpression of the same grammatical type, designating a concept or relation ofthe same logical 'level' as that designated by E. E and E' are assumed to beexpressions of the same Frege-Peano language. Lastly, we assume that theassociation of E with E' is one-one. Evidently the mapping E E' induces amapping G G' on sentences (and thus, on sets of sentences) when, as in thecase of a Frege-Peano (or a 'logically perfect') language, the syntax is appropriately specified.Frege asserts, as a 'basic law' [Grundgesetz] of dependence, that G isdependent upon g only if there is no such mapping E E' from expressions toexpressions, such that Q' is true and G' is false under the mapping:

Let us now consider whether a thought G is dependent upon a group ofthoughts Q. We can give a negative answer to this question if, according toour vocabulary, to the thoughts of the group g there corresponds a group oftrue thoughts Q ', while to the thought G there corresponds a false thoughtG'. For if G were dependent upon Q, then since the thOUghts of g ' are true,G' would also have to be dependent upon g ' and consequently G' would betrue.32

What suggests that Frege is here attempting an account of logical consequence isthe fact that he is explicitly concerned with cases where he must pass from thetruth of one proposition to a consideration of the logical possibility of the truthof another proposition, which may in fact be false. This is precisely what iseffected by the introduction of the mapping E ...... E'.Frege's presentation of his basic law of dependence follows an earlier, andmore characteristic account of logical dependence in terms of inferability:

Let Q be a group of true thoughts, Let a thought G follow from one orseveral of the thoughts of this group by means of logical inference such thatapart from the laws of logic, no proposition not belonging to g is used. Let usnow form a new group of thoughts by adding the thought G to the group C.Call what we have just performed a logical step. Now if through a sequence ofsuch steps, where every step takes the preceding one as its basis, we can reacha group of thoughts that contains the thought A, then we call A dependentupon the group g. 33By advancing his basic law of dependence as a law, Frege implicitlyacknowledges that he is not merely giving an elucidation of what was meant bythe earlier characterization in terms of inferability; otherwise he could not,consistently with the principles set forth in this series of essays, claim to haveformulated a law of dependence. However, I have found nothing in his

32 Frege 1984, p. 429. Frege passes freely between sentences and the thoughts or propositions theyexpress without an explicit change of notation. leaving it to the context to make clear wbich isintended. Our exposition follows him in this practice.33 Frege 1984, p. 334.
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Frege, Hilbert, and the Conceptual Structure o f Model Theory 223discussion to suggest that he considered the possibility that (Q, G) might satisfythe basic law, even though G is not logically inferable from Q , without, at thesame time, attributing this possibility to a limitation of the mapping, E ..... E' . Tothe extent that this is unclear, the attribution to Frege of a concept of logicalconsequence must remain somewhat tentative. Nevertheless, it should be notedthat exactly the same unclarity arises in the case of Bolzano, and Bolzano isroutinely accorded priority on this matter.34 By contrast, for Tarski there is nosuch unclarity: Tarski's account of logical consequence is explicitly designed toavoid precisely the limitations of analyses based on mappings like E ..... E' .We have argued that while there is in his discussion of dependence an echoof Bolzano and (a pre-echo) of Tarski circa 1936, Frege never achieved the keyelements of the model-theoretic explication of logical consequence: non-logicalconstants, structures, and the concept of truth in a structure. On our analysis,Frege failed to achieve these components of the model-theoretic picture as adirect result of the character of his concept-script and his theory of sense andreference; that theory was conceived for a language which, like the language ofthe Begriffsschrift, does not require the notions of relative reference and relativetruth, notions that are characteristic of Hilbert's anticipation of a modeltheoretic point of view. It should be noted that in arguing this thesis we have atno point maintained that Frege lacks a metatheoretical perspective. And indeed,that, in the essays on geometry, Frege had such a perspective is evident from hisformulation of his 'basic law' and his observation that

. .. to prove the independence of a real axiom from a group of real axioms. . . we [must] enter into a realm that is otherwise foreign to mathematics. Foralthough like all other disciplines mathematics, too, is carried out in thoughts,still thoughts are otherwise not the object of its investigations. Even theindependence of a thought from a group of thoughts is quite distinct from therelations otherwise investigated in mathematics. Now we may assume that thisnew realm has its own specifiC, basic truths which are as essential to theproofs constructed in it as the axioms of geometry are to the proofs ofgeometry; and that we also need these basic truths especially to prove theindependence of a thought from a group of thoughts.3S

Our account accommodates these facts while explaining Frege's failure to pursuethe development of a framework for treating metatheoretical questions alongmodel-theoretic or 'Hilbertian! lines. If we have diverged from earlier discussions of the Frege-Hil bert controversy, it is by locating its basis in differences intheir grasp of the conceptual structure of model theory-non-Iogical constants,structures, and truth in a structure-and, perhaps more importantly, byaccounting for Frege's failure to grasp this framework, not in terms of some vagueinability to deal with semantic or metatheoretic ideas, but in terms of the clash34 For example, Corcoran 1973, p. 71, says of BoIzano that he 'Indy deserves credit for (the)explication of logical consequence, if Tarsld does, beGause .. . Balzano offered precisely theidea.' But uide from the restriction to true premises, there is no difference betweenBolzano's characterization of deducibility and Frege's basic law of dependence. Bolzano'snotions of proposition-in-itself, idea-in-itself, and indeed, the whole tenor of his discussion, isquite dearly anticipatory of Frege.3S Frege 1984, p. 425. Among commentators, only Dummett 1976 appears to have perceived theimportance of the papers on foundations of geometry for our appreciation of Frege's grasp ofsuch issueS.

224 William Demopoulosbetween his theory of sense and his concept-script on the one side, and the'indexical' character of non-logical constants on the other.

AcknowledgementsI wish to thank John Corcoran, Michael Dummett, Michael Friedman,Michael Hallett, Howard Jackson, Michael Resnik, and Steven Wagner forcomments on earlier drafts. I am especially indebted to Mark Wilson for severalconversations on the topics dealt with in this paper. Financial support from theSocial Sciences and Humanities Research Council of Canada is gratefullyacknowledged.

ReferencesBoizano, Bernard 1837 Wissenscha/tslehre, (Sulzbach). Partial translations in Jan Berg,ed, Bernard Bolzano: theory of science, Dordrecht, (Reidel), 1973, tr. BurnhamTerrell, and in Bernard Bolzano: theory of science, ed. & tr. Rolf George, Blackwell(Oxford) 1972.Bernays, Paul 1942 Review of Em unhekannter Brief von Gotdob Frege aber Hilbertserste Vorlesung aber die Grundlo.gen der Geometrie, M. Steck, cd., Journal ofsymbolic logic 7, 92-93.Coffa, Alberto 1986 'From geometry to tolerance', in Ro bert Colodny, ed., From quarksto quasars, University of Pittsburgh Press. (Pittsburgh).Corcoran, John 1972/73 'Conceptual structure of classical logic', Phik;Jsophy andphenomenalogical research 33,25-47.- -1973 'The meanings of implication', Dialogos, 9, 59-76Corcoran, John and Susan Wood 1972/73 Review of Eike-Henner W. Kluge, ed. and tr.,

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Composito mathemotica 13,81-102. Reprinted in Tarski 1986, Vol. 3, pp. 653-74.van Heijenool1, Jan, ed. 1967 From Frege to Gi:ldel: a sourcebook in I1UIthel1Ulticallogic.Harvard University Press (Cambridge).Wilson, Mark 1992 'Frege: the royal road from geometry'. NoUs 26, 149-80.

Demopoulos - 1994 - Frege, Hilbert, And the Conceptual Structure of Model Theory

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