matization of physics,

Keywords:David Hilbert

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and later approached it as a 'vague' mathematical application in physics, he eventually understood

& 2015 Elsevier Ltd. All rights reserved.

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mathematical world (Corry, 2004, 3) (e.g. Corry, 1997; Corry,

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Studies in History and Philosophyof Modern Physics

e and ends with thepromises for the axiomatization of several physical disciplines in his 1905 lecture

lecture courses on mechanics and the kinetic theory and ends with his discussion

Studies in History and Philosophy of Modern Physics 53 (2016) 2844http://dx.doi.org/10.1016/j.shpsb.2015.11.004

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1 See, for example, Browder (1976), Corry (1997), Gray (2000a), (2000b), Reid(1996, Chapter 10), Wightman (1976) and Yandell (2002) for accounts of thebackground and inuence of Hilbert's lecture entitled Mathematical Problems(Hilbert, 1900 [2000]).

of theories of matter and electromagnetism. The third part begins with the com-pletion of the Foundations of Physics and ends with his lectures of 19211923. Thefourth one starts around 1922 and ends with Hilbert's joint paper on quantummechanics of 1928. Obviously, these parts of are not separated by clean-cutboundaries as, for instance, Corry (1999b) makes clear.1999a, 1999b, 1999c, 1999d, 2006c) the place of probability within course. Following the fth chapter of Corry (2004), entitled From mechanical toelectromagnetic reductionism: 19101914, the second part begins with Hilbert'spart; in the rst rank [is] the theory of probabilities' (Hilbert, 1900[2000], 418). The inclusion of, on the one hand, the axiomatizationof physics among the other 22 unsolved mathematical problemsand, on the other hand, the theory of probabilities among thephysical theories to be axiomatized, has long puzzled commenta-tors. Although it has been abundantly shown, in recent years, thatHilbert's [lifelong] interest in physics was an integral part of his

into four partially over-lapping sub-periods: 1902), 19101914 (Section 3), 19151923 (Section 4(Section 5).2 From the fact that each of these susponds to a specic position vis--vis, on thefoundations of physics and, on the other hand, p

2 The rst part begins with Hilbert's (1900) Paris lecturthose physical sciences in which mathematics plays an important and secondary literature available, this period can be separated01905 (SectionWhen citing this paper, please us

1. Introduction

It has become a commonplace toDavid Hilbert's (18621943) famouscentral starting point for modern ax(Hochkirchen, 1999; Schafer & VovkChapter 7).1 Here, Hilbert proposedfull journal title Studies in History and Philosophy of Modern Physics

o the sixth problem oflecture of 1900 as theized probability theory2006; Von Plato, 1994,t by means of axioms,

his own project of the axiomatization of physics has receivedcomparatively little explicit attention.

It is the aim of this paper to provide an account of the devel-opment of Hilbert's approach to probability between 1900 theyear in which called for the axiomatization of physics in his Parisaddress and 1928 the year in which he attempted to axioma-tize quantum mechanics. On the basis of the extensive primaryGeneral covariance

Structure of matterProbability theoryAxiomatization of physicsFoundations of modern science

probability, rst, as a feature of human thought and, then, as an implicitly dened concept without axed physical interpretation. It thus becomes possible to suggest that Hilbert came to question, from theearly 1920s on, the very possibility of achieving the goal of the axiomatization of probability as describedin the 'sixth problem' of 1900.The place of probability in Hilberts axioca. 19001928

Lukas M. VerburgtDepartment of Philosophy, University of Amsterdam, The Netherlands

a r t i c l e i n f o

Article history:Received 20 November 2013Received in revised form9 November 2015Accepted 24 November 2015

a b s t r a c t

Although it has become a cthe starting point for modecomparatively little explicithis topic in light of the cezation of physics went hameaning of probability. Whmon place to refer to the 'sixth problem' of Hilbert's (1900) Paris lecture asxiomatized probability theory, his own views on probability have receivedtention. The central aim of this paper is to provide a detailed account ofal observation that the development of Hilbert's project of the axiomati-in-hand with a redenition of the status of probability theory and theHilbert rst regarded the theory as a mathematizable physical discipline

2005; Stachel, 1999, see also Earman & Glymour, 1978; Vizgin,2001). Hilbert's philosophical reections, of the early 1920s, on histheory dealt with the epistemological implications of generalcovariance such as time-reversal invariance and new conditionsfor the objectivity and completeness of physical theories based ongeneral relativity and quantum mechanics. Where probability was

minimal set of [. . .] axioms' (Hilbert, 18981899 quoted in Corry,5

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 2844 29follows the paper's main observation; namely, that there was afundamental change in Hilbert's approach to probability in theperiod 19001928 one which suggests that Hilbert himselfeventually came to question the very possibility of achieving thegoal of the mathematization of probability in the way described inthe famous sixth problem (Mathematical treatment of theaxioms of physics). In brief, Hilbert understood probability, rstly,as a mathematizable and axiomatizable branch of physics (19001905), secondly, as a vague statistical mathematical tool for theatomistic-inspired reduction of all physical disciplines tomechanics (19101914), thirdly, as an unaxiomatizable theoryattached to the subjective and anthropomorphic part of the fun-damental laws for the electrodynamical reduction of physics(19151923) and, fourthly, as a physical concept associated tomechanical quantities that is to be implicitly dened through theaxioms for quantum mechanics (19221928). Because Hilberttended not to stress the state of ux, criticism, and improvement(Corry, 2004, 332) that his deepest thoughts about physical andmathematical issues were often in there inevitably is a certainamount of speculation involved in connecting these four sub-periods. Consequently, what follows is to be considered as onepossible way of accounting for Hilbert's remarkable change ofmind in the period 19001928.

1.1. Overview of the argument

(1) Between the years 1900 and 1905, Hilbert proposed notonly to extend his axiomatic treatment of geometry to the physicaltheory of probability, but also to let this treatment be accompaniedby the further development of its (inverse) applications inmathematical-physical disciplines (Corry, 2006c). On the onehand, given that an axiomatization is to be carried out retro-spectively, the suggestion that geometry was to serve as a modelfor the axiomatization of probability implied that Hilbert thoughtof the theory as a more or less well-established scientic dis-cipline. On the other hand, the fact that the statistical method ofmean values for the kinetic theory of gases was to be rigorized bymeans of probability theory's axiomatization pointed, rstly, to theunsettled status of probabilistic methods in physics and, secondly,to the possibility of having the axiomatic method restore it. (2) Theyears 19101914 could be separated into three phases. Firstly, from19101912/1913, Hilbert explicitly elaborated the atomistichypothesis as a possible ground for a reductionist mechanicalfoundation for the whole of physics in the context of severalphysical topics based on it (Corry, 1997,, 1998,, 1999d,, 2000,, 2004,,2006a). It was under the inuence of his increasing acknowl-edgment of the disturbing role of probabilistic methods (e.g.averaging) in the mathematical difculties involved in the axio-matization of physics in general and [the] kinetic theory in par-ticular (Corry, 2004, 239) from the atomistic hypothesis that Hil-bert, secondly, became more and more interested (from late-1911/early-1912 on) in coming to terms with these difculties via aninvestigation into the structure of matter (Corry, 1999a, 1999d,2010). Thirdly, as a result of his consideration of the mathematicalfoundations of physics, in the sense of radiation and moleculartheory as going beyond the kinetic model as far as its degree ofmathematical sophisticated and exactitude is concerned (Corry,2004, 237), Hilbert eventually came to uphold Mie's electro-dynamical theory of matter by late-1913/early-1914 (Corry, 1999b;Mehra, 1973; Section 3.4, see also Battimelli, 2005; McCormmach,1970). (3) The third period (19151923) pivots around theappearance, in 1915, of the Foundations of Physics in which Hil-bert presented a unied eld theory, based on an electrodynamicalreductionism, that combined Mie's theory and Einstein's (non-covariant) theory of gravitation and general relativity (e.g. Corry,

2004; Majer & Sauer, 2005; Renn & Stachel, 1999; Sauer, 1999,2004, 90) , in mechanics

all physicists recognize its most basic facts [but] the arrange-ment [is] still subject to a change in perception [and] thereforemechanics cannot yet be [turned into] a pure mathematicaldiscipline, at least to the same extent that geometry is (Hilbert,18981899 quoted in Corry, 2004, 90).

3 In 1894 Hilbert gave a lecture course on The Foundations of Geometry.4 Hilbert made this remark at several occassions between 1893 and 1927 - for

example in his 1898/1899 lecture entitled 'The Foundations of Euclidean Geometry'.The only place where Hilbert did not stress this aspect of his position was in hisGrundlagen der Geometrie.

5 In the year 1898-1899 Hilbert gave his rst full course on a physical topic,here accepted as a subjective accessorial principle implied in theapplication of the laws of the new modern physics to nature, (4) inhis later contributions to the axiomatization of quantummechanics empirical probabilities were implicitly dened throughthe axioms of a yet uninterpreted formalism after physicalrequirements had been put upon them (e.g. Lacki, 2000).

2. First period. The axiomatization of probability as a physicaldiscipline: 19001905

Hilbert's Grundlagen der Geometrie of 1899 resulted from hisattempt to lay down a simple and complete system of independentaxioms for the undened objects points, lines and planes thatestablish the mutual relations that these objects are to satisfy(Hilbert, 1899 [1902], see also Hilbert, 1891 [2004],, 1894 [2004],Toepell, 1986b). In the lecture notes to a course on the Founda-tions of Geometry of 1894, Hilbert dened the task of the appli-cation of the axiomatic method to geometry as one of determining

the necessary, sufcient, and mutually independent conditionsthat must be postulated for a system of things, in order that anyof their properties correspond to a geometrical fact and, con-versely, in order that a complete description and arrangementof all the geometrical facts be possible by means of this systemof things (Hilbert, 1894 3 quoted in Toepell, 1986a, 5859, myemphasis).

Hilbert was of the opinion that his axiomatization of elemen-tary geometry was part of a more general program of axiomati-zation for all of natural science (e.g. Majer, 1995) and that geo-metry, as the science of the properties of space, must be con-sidered as the most perfect of the natural sciences4 see Toepell,1986a, vii, see also Corry, 2006b; Majer, 2001). The fact that Hil-bert's axioms for geometry were chosen so as to reect spatialintuition not only indicates that the axiomatic method itself is atool for the retrospective, or post-hoc, investigation of the logicalstructure of concrete, well-established and elaborated [. . .] enti-ties (Corry, 2004, 99). But it also suggests that the differencebetween geometry and, for example, mechanics pertained solelyto the historical stage of the development of both sciences. Wherethe basic facts of geometry are so irrefutably and so generallyacknowledged [that] no further proof of them is deemed necessary[and] all that is needed is to derive [the] foundations from anamely mechanics.

Hilbert's motivation for the application of the axiomaticmethod to physical theories was to create a secure foundationfrom which all its known theorems could be deduced so as toavoid the recurrent situation in which new hypotheses intended toexplain newly discovered phenomena are added to existing the-ories without showing that the former do not contradict the latter.It is against this background that Hilbert's inclusion of the axio-matization of physics in his 1900 Paris lecture must be understood.

2.1. The place of probability in the sixth problem of Hilbert's (1900)Paris lecture

Hermann Minkowski (1864-1909) who both saw clearly that the

When we are engaged in investigation the foundations of ascience, we must set up a system of axioms which contains anexact and complete description of the relations subsistingbetween the elementary ideas of that science. The axioms soset up are at the same time the denitions of those elementaryideas; and no statement within the realm of the science whosefoundations we are testing is held to be correct unless it can bederived from those axioms by means of a nite number oflogical steps (Hilbert, 1900 [2000], 414).

The sixth problem itself was described in terms of the call forthe treatment in the same manner, by means of axioms, [of] thosephysical sciences in which mathematics plays an important part, inthe rst rank [. . .] the theory of probabilities and mechanics(Hilbert, 1900 [2000], 418). After having mentioned the work of, onthe one hand, Ernst Mach (18381915), Ludwig Boltzmann (18441906), Paul Volkmann (18561938) and Heinrich Hertz (18571894) on mechanics and, on the other hand, Georg Bohlmann(18691928) on the calculus of probabilities as important con-tributions to these branches of physical science, Hilbert wrote that

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 284430publication of the Foundations of Geometry had opened up animmeasurable eld of mathematical investigation [. . .] which goesfar beyond the domain of geometry (Hurwitz quoted in Gray,2000a, 2000b, 10) Hilbert proposed that his mathematics ofaxioms (Hurwitz quoted in Gray, 2000a, 2000b, 10) should beapplied to both the empirical and pure part of mathematics6:

I [Hilbert] [. . .] oppose the opinion that only the concepts ofanalysis [. . .] are susceptible of a fully rigorous treatment [forthis] would soon lead to the ignoring of all concepts arising [forinstance] from [mathematical] physics, to a stoppage of theow of new material from the outside world. [W]hat animportant nerve, vital to mathematical science, would be cut by[its] extirpation! [I] think that wherever, from the side of [. . .]geometry, or from the theories of [. . .] physical science,mathematical ideas come up, the problem arises [. . .] toinvestigate the principles underlying these ideas and so toestablish them upon a simple and complete system of axioms,that the exactness of the new ideas and their applicability todeduction shall [not] be [. . .] inferior to those of the oldarithmetical concepts (Hilbert, 1900 [2000], 410).

2.1.1. The sixth problemIf Hilbert introduced the sixth problem (The mathematical

treatment of the axioms of physics) by writing that it is suggestedby the axiomatic investigations on the foundations of geometry(Hilbert, 1900 [2000], 418), the structure of this investigation thatwas to serve as a model (Hilbert, 1900 [2000], 419) for its solutionwas summarized as part of the second problem (The compatibilityof the arithmetical axioms) on the list:

6 Before presenting the problems, Hilbert made clear that it is the interaction,or dialectical interplay (Corry, 1997, 65), between the empirical part (i.e. thenatural sciences) and pure part (e.g. the theory of numbers and functions andalgebra) of mathematics that accounts for its organic development: [T]he rst andoldest problems in every branch of mathematics spring from experience and aresuggested by the world of external phenomena [I]n the further development of abranch of mathematics, the human mind, encouraged by the success of its solu-tions, becomes conscious of its independence. It evolves from itself alone [. . .] bymeans of logical combination, generalization, specialization, by separating andcollecting ideas in fortunate ways, new and fruitful problems, and appears thenitself as the real questioner [. . .] In the meantime, while the creative power of purereason is at work, the outer world again comes into play, forces upon us newquestions from actual experience [and] opens up new branches of mathematicsThe central goal of Hilbert's invited lecture at the SecondInternational Congress of Mathematicians, held in Paris, was to liftthe veil behind which the future [of mathematics] lies hidden(Hilbert, 1900 [2000], 407) by formulating, for its gifted mastersand many zealous and enthusiastic disciples (Hilbert, 1900 [2000],436), 23 problems which the science of today sets and whosesolution we expect from the future (Hilbert, 1900 [2000], 407).Following the advice of his friends Adolf Hurwitz (18591919) and(Hilbert, 1900 [2000], 409).[i]f geometry [serves] as a model for the treatment of physicalaxioms, we shall try rst by a small number of axioms toinclude as large a class as possible of physical phenomena, andthen by adjoining new axioms to arrive gradually at the morespecial theories (Hilbert, 1900 [2000], 419).7

Although the treatment of physics along the lines of an axio-matic method that was rst applied to (Euclidean) geometry ttednaturally into Hilbert's early views on the connection between(pure) mathematics, geometry and physics and, in fact, had beenpart of the evolution of Hilbert's early axiomatic conception(Corry, 1997, 70) , Hilbert's inclusion of the sixth problem on thelist was remarkable. Firstly, in so far as it neither satises his owncriteria as to what a meaningful problem in mathematics is nor sitswell with his statement that a solution to a mathematical problemshould be obtained in a nite number of steps,8 it is, secondly,better understood as a general task or research program. Andthirdly, if it is difcult to understand the place of this project aspart of [Hilbert's] work [on mathematical physics] up to that time[. . .], unlike most of the other [problems] in the list, [it] is[fourthly] not the kind of issue that mainstream mathematicalresearch had been pointing to in past years (Corry, 1997, 69). Thefact that, especially, the mentioning of probability theory as one ofthe main physical sciences that are amenable to axiomatization

7 Hilbert added that [t]he mathematician will have also to take account notonly of those theories coming near to reality, but also, as in geometry, of all logicallypossible theories. He must be always alert to obtain a complete survey of all con-clusions derivable from the system of axioms assumed (Hilbert, 1900 [2000], 419).About this statement Corry (1997, 89) has made the following lucid remark:[Hilbert thus] suggested that from a strictly mathematical point of view, it wouldbe possible to conceive interesting systems of physical axioms that [. . .] dene akind of non-Archimedean physics (Corry, 1997, 89). He did not consider suchsystems here, however, since the task was to see how the ideas and methods ofaxiomatic can be fruitfully applied to physics [. . .] [And] [w]hen speaking ofapplying axiomatic ideas and methods to these theories, Hilbert meant [. . .] existingphysical theories. But [an important] possibility [was] suggested.

8 In his Mathematical Problems, Hilbert wrote the following: It remains todiscuss [. . .] what general requirements may be justly laid down for the solution ofa mathematical problem. I should say [that] it shall be possible to establish thecorrectness of the solution by means of a nite number of steps based upon a nitenumber of hypotheseswhich are implied in the statement of the problem and whichmust always be exactly formulated. This requirement of logical deduction by meansof a nite number of processes is simply the requirement of rigor in reasoning. [Also][t]he conviction of the solvability of every mathematical problem is a powerfulincentive to the worker [. . .] There is the problem. Seek its solution. You can nd itby pure reason, for in mathematics there is no ignorabimus' (Hilbert, 1900 [2000],409 and 412). This last statement must be read as a response to Emil du Bois-Reymond's (18181896) ignoramus et ignorabimus (we do not know and will not

know) (e.g. McCarty, 2005).

has often puzzled readers (Corry, 1997, 68), suggests that, apartfrom these four general features of the sixth problem, there areother more specic features peculiar to Hilbert's treatment of thistheory.

2.1.2. Hilbert, Bohlmann and probability theoryIn his Paris lecture, Hilbert referred to Bohlmanns (1900) ber

Versicherungsmathematik for a presentation of the axioms of the

envisaged as the solution [. . .] of the sixth of his 1900 list ofproblems (Corry, 1997, 73). But Hilbert himself cautioned not onlythat the completion of the axiomatization of physics is to beunderstood as a research program (Arbeitsprogramm), but alsothat in the individual physical disciplines mentioned it is merelypossible to nd initiatives which only in some cases have beencarried through (Hilbert, 1905 quoted in Corry, 1997, 84).12 Afterdiscussing Bohlmann's, 1901 article (Section 2.2.1), Hilbert alsointroduced three applications of probability: The theory of com-pensation of errors, the kinetic theory of gases and insurancemathematics (Section 2.2.2).

2.2.1. Early axiomatic systems of probability: Hilbert and BohlmannThe axioms for probability theory that Hilbert presented were

drawn from Bohlmann's (1901) lengthy article Lebensversicher-ungsmathematik which contained a much more precisemathematical formulation of the axioms underlying the mathe-matical treatment of life insurance, which in [his] earlier article [of1900] appear[ed] as very general, somewhat loosely formulatedassumptions (Corry, 1997, 129). After formulating his main claim,

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 2844 31theory of probabilities (Hilbert, 1900 [2000], 418). This paperreproduced several talks which Bohlmann delivered during Easterof the year 1900 in which he announced that his much morerigorous axiomatization of probability would appear in an articleon life-insurance mathematics (Lebensversicherungsmathematik)(Bohlmann, 1901) for Felix Klein's (18491925) Encyklopdie dermathematischen Wissenschaften.9 Given that Hilbert, who by theend of March 1900 had not yet determined the subject of his Paristalk [and] had not produced a lecture [.] [b]y June (Krengel, 2011,5, see also Reid, 1996, 70), had not shown interest in probabilitytheory prior to those months it seems certain that the problem ofits axiomatization was proposed to him by his colleague at Gt-tingen, Bohlmann.10

Hilbert made no secret about the fact that he had solicitedvarious problems by talking to other mathematicians (Krengel,2011, 5), but in light of the words that Bohlmann himself used tocharacterize his 1900 article (very general, loosely formulated), itis surprising that Hilbert decided to insist on its axiomatization. Forit may be recalled that Hilbert dened the axiomatic method as apost-hoc reection on the results of well-established physical dis-ciplines. Two other remarks can be made about Hilbert's descriptionof probability as a physical science in which mathematics plays animportant part. Firstly, in so far as Bohlmann's article did not pointout with which physical phenomena the theory is concerned, itremained unclear how the rst step of the treatment of its axioms,namely the inclusion of as large a class as possible of physicalphenomena [by] a small number of axioms (Hilbert, 1900 [2000],419), was to be carried out. Secondly, if Hilbert was convinced thatBohlmann had already put forward the axioms of the physicaldiscipline of probability, he emphasized that their logical investi-gation (Hilbert, 1900 [2000], 418) was also to be instrumental tosecuring the role of the so-called method of mean values inmathematical physics, and in particular in the kinetic theory ofgases (Hilbert, 1900 [2000], 418). These three points that of theissue of its doubtful status as a well-established and physical scien-tic discipline and that of the two-sided importance of its axio-matization reappeared in Hilbert's lecture course of 1905 entitledLogical Principles of Mathematical Thinking (Logische Prinzipiendes mathematischen Denken).

2.2. Probability as an application in (mathematical) physics

In the second part of this course Hilbert applied his axiomaticmethod to several physical disciplines (Hilbert, 1905; Corry, 1997,Section 8; Corry, 2004, Chapter 3);11 mechanics, thermodynamics,probability calculus, the kinetic theory of gases, insurancemathematics, electrodynamics and psychophysics. This can,indeed, be considered the rst clear evidence of what [he]

9 In a footnote to his 1900 article, Bohlmann remarked that the 1901 articlewould contain a much more precise, or rigorous, mathematical formulation of theaxioms of his treatment of life insurance.

10 Bohlmann was the rst lecturer of the new German institute of insurancescience (Sterbens-wahrscheinlichkeit) at Gttingen where he gave courses onactuarial mathematics until 1901, the year in which he became extraordinaryprofessor of actuarial mathematics.

11 The manuscript of this part of the lecture course has, unfortunately, not been

published.namely that the mathematical foundations of life insuranceground probability theory,13 Bohlmann referring to Poincar(1896) Calculdes Probabilits (with the telling subtitle Cours dephysique mathematique) as the main source of his axiomatization described probability in an axiomatic way, rstly, by distin-guishing between the general axioms of probability and the spe-cic axioms of life insurance and, secondly, by introducing threedenitions and axioms for the general calculus of probability (derallgemeinen Wahrscheinlichkeitsrechnung).14 In brief, Bohlmanndened the probability of an event E as a nonnegative numberp E 0rpEr1, for which it holds that (1) if E is the certainevent, then p E 1, if E is impossible, then p E 0, and (2) if E1and E2 happen simultaneously with probability zero, then theprobability of either E1 or E2 equals p E1 pE2. Bohlmann wasthe rst to treat probability as a function of an event,15 but hissystem contained several aws; the notion of an event Eremained undened, the requirement that pE is rational wasalready outdated and denitions and axioms [. . .] appear inter-mingled in a way that Hilbert himself would have avoided if he[would have] systematically followed the model of the ['Founda-tions of Physics'] (Corry, 1997, 130).

However, Hilbert merely restated Bohlmann's (1) and (2) usinga slightly different notation.16 He did not comment on the

12 Anstze dazu, die nur in ganz wenigen Fllen durchgefhrt sind (Hilbert,1905 quoted in Corry, 1997, 84).

13 Bohlmann mentioned Karl Wagner (?-?), who, in his monograph entitledDas Problem von Risiko in der Lebensversicherung of 1898, had argued that prob-ability theory and insurance are, in essence, uncon-nected, as an opponent of thisview (Koch, 1998, Section 2.2; Purkert, 2002).

14 Bohlmann credited the rst group of axioms to Emanuel Czuber (18511925)(Czuber, 1900) and the second group to Ladislaus von Bortkiewicz (18681931)(Von Bortkiewicz, 1901).

15 There is some controversy as to the historical and theoretical importance ofBohlmann's contribution to probability theory. Where Von Plato writes that Bohl-mann does not do much more than call some of the basic properties of probabilitycalculus by the name of axioms (Von Plato, 1994, 32), Krengel criticizes Von Platofor having little feeling for the magnitude of [the] step [of dening probability as afunction of an event, LV], which is even more evident if one thinks about the longtime it took for this idea to be accepted (Krengel, 2011, 4). See also Hochkirchen(1999, 2831) and Schneider (1988, 509) for short discussions of Bohlmann.

16 The simultaneous occurrence of two events E1, E2 is [. . .] denoted by E1E2,whereas E1 E2 denotes their disjunction. Two events are mutually exclusive ifp E1E2 0, while pE1jE2, denotes conditional probability (Hilbert quoted inCorry, 2004, 165). The following two axioms were put forward as the denition ofprobability theory: I. p E1 E2 p E1 pE2, if p E1E2 0. II.p E1E2 pE1 p E1E2 (Corry, 2004, 165). As to Hilbert's presentation of Bohl-mann's axiomatization of probability theory, Corry (2004, 165) also remarks that

Hilbert did not mention an additional denition appearing in Bohlmann's article,

independence, consistency or completeness of these axioms [suchthat his] system was a rather crude one by [his] own criteria(Corry, 1997, 132). There are four remarks to be made about thissituation. Firstly, Hilbert's exible treatment of the axiomatizationof probability was at odds with the position occupied in his cor-respondence of 18971902 with Gottlob Frege (18481925) on theaxiomatic method (Blanchette, 1996; Demopoulos, 1994; Hallett,2010; Resnik, 1974). In a letter of 22 September, 1900 Hilbert wrotethat a concept can only be logically xed through its relations toother concepts. These relations, formulated in denite statements,I call axioms, and thus I arrive at the view that these axioms [. . .]are the denitions of these concepts (Hilbert quoted in Hallett,2010, 427). If it seems likely, secondly, that Hilbert, [i]n treatingthe axioms of probability and speaking of the need to separate rather than to combine axioms and denitions, [was] stressingthe early state in which the theory was then found (Corry, 1997,131), it may also be emphasized, thirdly, that more sophisticatedattempts at the axiomatization of probability had already beenmade since Bohlmann's (1901) article. It was Rudolf Laemmel

2.2.2. The applications of probability theory in Hilberts (1905) lec-ture course

The topic of the application of probability had already appearedas an aspect of the sixth problem of the Paris lecture. Here, Hil-bert wrote that it seems [. . .] desirable that the logical investi-gation [of] the axioms of the theory of probabilities [is] accom-panied by a [. . .] satisfactory [treatment] of the method of meanvalues in mathematical physics, and in particular the kinetic theoryof gases (Hilbert, 1900 [2000], 418, my emphasis). Perhapsunsurprisingly, next to Sterbenswahrscheinlichkeit (insurancemathematics),22 the 1905 lecture course presented Ausgle-ichungsrechnung (method of mean values, or theory of com-pensation of errors (TCE)) (Corry, 1997, 133136, see also Aldrich,2011; Sheynin, 1972,, 1979) and the kinetische Gastheorie (kinetictheory of gases (KTG)) (Corry, 1997, 136152, see also Brush, 1976;Brush & Hall, 2003; Loeb, 1961) as the main applications ofprobability theory. Hilbert wanted to nd a mathematical basis for

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 284432(18791962) who, in his dissertation of 1904 entitled Untersu-chungen ber die Ermittlung von Wahrscheinlichkeiten,17 for-mulated, in set-theoretical terms, two axioms (of total and com-pound probability) and three denitions. After having establishedtheir independence and sufciency, Laemmel wrote that theaxioms allowed for the hypothetical18 construction of a systemfor probability.19 Laemmel did not refer to Bohlmann's articles andbecause it is unclear whether he was acquainted with Hilbert'sFoundations of Geometry it cannot be said that he intended,through his axiomatization, to solve the sixth problem (Von Plato,1994, 33). If Hilbert, for his part, was not familiar with Laemmel'swork, he did not seem to be very interested in the doctoral dis-sertation of his student Ugo Broggi (18801965) of 1907 entitledDie Axiome der Wahrscheinlichkeitsrechnung in which he fol-lowing the criteria of Hilbert's axiomatic method set out toperfect the earlier proposals of Bohlmann and Laemmel.20,21 Not-withstanding his own acknowledgment of the early state ofprobability theory, Hilbert, thus, did neither improve upon Bohl-mann's account nor involve himself with the novel attempts ofothers to axiomatize the theory along the lines of his own fra-mework. This indicates, fourthly, that Hilbert, in 1905 [. . .] wasmuch less interested in the calculus of probabilities as such (Corry,1997, 133) let alone as a physical discipline than in its[mathematical] applications (Corry, 1997, 133).

(footnote continued)namely, that two events E1, E2 are independent if the probability of their simul-taneous occurrence equals p E1 pE2 (Corry, 2004, 165).

17 For excerpts of Laemmel's dissertation see Schneider (1988, 359366) andfor brief discussions Corry (1997, 132), (2004, 166), Hochkirchen (1999, 137),Schafer & Vovk (2006, 83) and Von Plato (1994, 3233).

18 Laemmel was of the opinion that in order to ascertain to probabilities agreater epistemological value, one has to try to replace the intuitive or empiricalprocedure by a determination of probability through a process of hypotheses(Laemmel 1904 quoted in Von Plato, 1994, 32).

19 It may be noted that Laemmel left the problem of consistency untouchedand did not explicate the concept of independence.

20 Hilbert did not notice the falsehood of Broggi's proof of the claim thatdenumerable or countable additivity can be derived from nite additivity. Thismistake was later exposed by another student of Hilbert, Hugo Steinhaus (18871972) in his Les probabilits dnombrables et leur rapport la thorie de lamesure of 1923 (Steinhaus, 1923, see also Hochkirchen, 1999, 236; Schafer & Vovk,2006, 83, 8586).

21 After proposing a system of axioms for probability, Broggi showed thecompleteness, mutual independence and consistency of his axioms. His rst axiomstated that the certain event has value 1 and the second axiom consisted of the ruleof total probability. Broggi dened probability as a ratio (in a discrete or geometric)setting and then veried his axioms. If Laemmel and Broggi's proposals can be

distinguished from the system of Bohlmann with reference to the fact that theythe determination of average values which, for example in thecontributions of James Clerk Maxwell (18311879) (Brush, 1958;Dias, 1994; Garber, 1970; Garber, Brush, & Everitt, 1986) and Lud-wig Boltzmann (1844-1906) (Bernstein, 1963; De Regt, 1999;Wilholt, 2002) to the KTG, were determined with the help of a[so-called] probability distribution for the quantity considered(Von Plato, 1944, 3334, see also Garber, 1973; Krajewski, 1974;Von Plato, 1994, Chapter 3) (see Section 3.1).23

It was under the inuence of the views put forward in themathematical physics seminar of Franz Neumann (17981895) andJacob Jacobi (18041851) at the University of Knigsberg (Olesko,1991) that Hilbert had become convinced of the value of theexactness of measurement in physical research.24 He rst claimedthat the whole TCE could be derived from the axiom that if var-ious values have been obtained from measuring a certain magni-tude, the most probable actual value of the magnitude is given bythe arithmetical average of the various measurements (Hilbert1905 quoted in Corry, 1997, 134). Referring to an article of theastronomer Julius Bauschinger (18601934), Hilbert, then, gaveGauss's error theory and the method or principle of least squaresas its two theorems. Because the axiom and the two theoremswere entirely equivalent (vollkommen aequivalent) i.e. any oneof them could be deduced as a mathematical consequence fromthe other two it was arbitrary which one was taken as thefoundation of the theory.25 The new work that was to be expectedin this domain was to aim for the (axiomatic) reduction of thesethree statements to other axioms with a more limited content andgreater intuitive plausibility (Corry, 1997, 135).

It is generally agreed that probability rst entered (statistical)physics in the work of Rudolf Clausius (18221888) and Karl

(footnote continued)start from set theory, Broggi moved beyond Laemmel by making use of measuretheory as well.

22 It may come as no surprise that Hilbert, also here, followed Bohlmann intaking the abovementioned axioms of probability and adding more specic de-nitions and axioms (see Corry, 1997, 152154).

23 In Section 3 it will be made clear that this endeavor became of centralimportance for his search for, and eventual commitment to, a specic theory of thestructure of matter that could serve as the basis for the foundation of physics.

24 Hilbert remained at the University of Knigsberg between 1880 and 1895. Itis likely that he participated in the seminar and, perhaps, even attended the lec-tures of Franz Neumann who, despite his retirement in 1876, was still to be seenat university gatherings and sometimes still lectured (Reid, 1996, 9) after thatparticular year.

25 Hilbert wrote that what one [. . .] really [. . .] wants to consider as thefoundation when there are several possibilities, is also here arbitrary and depen-dent upon personal inclinations and the general state of science ([W]as man [. . .]wirklich [. . .] als Grundlage ansprechen will, wenn sich so verschie-deneMglichkeiten ergeben haben, is wie stets willkrlich und hngt von persnli-chen Momenten und dem allgemeinen Stande der Wissenschaft ab) (Hilbert

quoted in Corry, 1997, 135).

Krnig (18221879) (e.g. Garber, 1973) the basis of which wasthe idea that even though [t]he path of each gas atom [is] veryirregular, so that it eludes calculation [. . .] according to the laws ofthe probability calculus one may assume, instead of this perfectirregularity, a perfect regularity (Krnig quoted in Schneider, 1988,300) on average. In a paper of 1860, Maxwell combined a revision

It is important to emphasize that where in 1900 the project ofthe axiomatization of probability theory was said to have to beaccompanied by a further development of its applications, thisparticular topic came to determine the core of Hilbert's (instru-mental) denition of probability theory after 1905. Rather thanaiming for the axiomatization of the theory per se, Hilbert thus

things, employed new (non-probabilistic) mathematics, he even-

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 2844 33of Clausius concept of the mean free path of a molecule with JohnHerschel's (17921871) treatment of the normal distribution oferrors to derive his distribution law for molecular velocities.Maxwell wrote that in order to calculate most of the observableproperties of a gas it is not necessary to know the positions andvelocities of all particles at a given time: It sufces to know theaverage number of molecules having various positions and velo-cities (Corry, 1997, 137). Similar to Boltzmann, Maxwell set out toexplain the behavior of macroscopic matter in terms of statisticallaws describing the motion of the atoms which themselves obeyNewtons mechanical laws of motion.

Hilbert noted that this development had made it clear not onlythat probabilistic assumptions had to be introduced into thedescription of physical systems, but also that probability theorywas required for the mechanical treatment of (heat) phenomena. Ifhe praised the theory for the remarkable way in which [it] com-bined the postulation of far-reaching assumptions about thestructure of matter [. . .] with the use of [both] probability [andinnitesimal] calculus (Corry, 2004, 169) to produce new physicalresults, Hilbert also cautioned that since probability theory is notan exact mathematical discipline (ist keine exakte mathematischeTheorie) it may only be used as a rst orientation (zu einer erstenOrientierung) and this when its results are correct and inaccordance with the facts of experience or with the acceptedmathematical theories (Hilbert, 1905 quoted in Corry, 2004, 170171, f. 158).26 The principal reason for this was that probabilitycalculations could produce results that were in conict with theaccepted laws of mechanics or, for that matter, with the obser-vable phenomena described by (partial) differential equations.More in specic, Hilbert observed that the abovementioned pro-visos were not always satised in the case of the then current KTG.For example, the application of probability could lead to (falla-cious) conclusions that contradicted other well-known results ofthe KTG, some of the results of the probabilistic version of the KTGconicted with the reversibility of the laws of mechanics, andsome of the proofs appearing in the laws of the KTG, such as thosepertaining to average velocities instead of to velocity distributionsas they actually occur, are in conict with the idea of the reduc-tion of phenomena to mechanical interactions between rigid,atomic, particles. Hilbert

wished to undertake an axiomatic treatment of the [KTG] notonly because it combined physical hypotheses with probabil-istic reasoning in a scientically fruitful way [but] also because[it] was a good example of a physical theory where [. . .]additional assumptions had been gradually added to existingknowledge without properly checking the possible [. . .] dif-culties that would arise from this addition. [For example] thequestion of probability in physics was not settled in this con-text (Corry, 2004, 168).

26 In Hilbert's own words, probability theory ist keine exakte mathematischeTheorie, aber zur einer ersten Orientierung, wenn man nur all unmittelbar leichtersichtlichen mathematische Tatsache benutzt, hug sehr geeignet; sonst fhrt siesofart zu grossen Verstssen. Am besten kann man immer nachtrglich sagen, dassdie Anwendung der Wahrscheinlichkeit immer dann berichtigt und erlaubt ist, wosie zu richtigen, mit der Erfahrung bzw. Der sonstigen mathematischen Theoriebereinstimmenden Resultaten fhrt (Hilbert, 1905 quoted in Corry, 2004, 170

171, f. 158).tually abandoned mechanical reductionism as a foundationalassumption as such. The electromagnetic reductionism that Hil-bert adopted around late-1913/early-1914 left open the question ofapplying probability principles in the KTG and left unanswered theproblems of introducing probability concepts in statisticalmechanics. However, Hilbert's approach to probability underwentan importance change under inuence of his gradual adoption ofthis new form of reductionism. On the one hand, there was the so-called physical viewpoint, articulated as part of the search for atheory of the structure of matter that would deepen the atomismof the kinetic theory (e.g. Wilholt, 2002), that Hilbert put forwardto overcome the mathematical difculties implied by the atomistichypothesis and that would manifest itself in [his] reconsiderationof his view of mechanics as the ultimate explanation of physicalphenomena (Corry, 1999a, 14) (Section 3). On the other hand, theviewpoint was an early suggestion of both the fact that and theway in which his attempt to come to terms with Einstein's newrelativistic mechanics via electromagnetism also meant that thestatus of probability, rather than that of the methods of probabilitytheory, became a central concern for Hilbert (Section 4).

3. Second period. Beyond mechanical reductionism beyondprobability theory: 19101914

After the 1905 lecture course, Hilbert lectured on mechanics(WS 1905/1906), continuum mechanics (SS 1906 and wintersemester of 1906/1907), differential equations of mechanics (WS1907/1908) and electrodynamics (SS 1907) (see Sauer & Majer,2009, 709726; Corry, 2004, 450452). Although he gave nocourses on physics until 1910, Hilbert followed his friend andcolleague at Gttingen Minkowski's implementation of the projectof the axiomatization of physics in the context of the investigationof the role of the new principle of relativity in various physicaltheories (see Corry, 2004, Chapter 4; Corry, 2010; Galison, 1979;Walter, 1999; Walter, 2008).28 Hilbert resumed his physical

27 Corry writes that [a]lready in his 1905 lectures on the axiomatization ofphysics Hilbert had stressed the problems implied by the combined application ofanalysis and the calculus of probabilities as the basis for the kinetic theory, anapplication which is not fully justied on mathematical grounds. In his physicalcourses after 1910 [he] expressed again similar concerns. Yet, the more Hilbertbecame involved with the study of the kinetic theory itself [. . .] these concerns diddevoted himself to the axiomatization of the KTG and, therefore,the TCE as its main theoretical applications.

Hilbert admired the KTG for the way in which this theorycombined the postulation of far-reaching assumptions about the[atomistic] structure of matter with the use of probability calculus(Corry, 2004, 168), but he increasingly came to recognize not onlythat the use of probabilities in the KTG and in physics at large wasstill in need of justication.27 But also that the foundationalquestions involved in the attempt to justify the introduction of notyet fully developed (Hilbert 19111912 quoted in Schirrmacher,2003, 10) probabilistic methods in physical theories requiredfurther investigation into the theory of matter as such (Corry,2004, 284). Where Hilbert initially searched for another theory ofmatter that could function as the basis for a reductionistmechanical foundation of physics in so far as it, among othernot diminish. Rather, they increased (Corry, 1999a, 13).

lectures in 1910 with a course on mechanics (WS 1910/1911) andcontinued to expand the scope of his interest to topics such ascontinuum mechanics (kinetic theory of gases) (SS 1911), statis-tical mechanics (winter semester 1911/1912), radiation theory (SS1912) and electromagnetic oscillations (WS 1913/1914) in yearsthat followed. The period between 1910 and 1914 was character-ized not only by a continual commitment to mechanical reduc-tionism, but also by a gradual, albeit implicit, endorsement, fromthe end of 1913 on, of electromagnetic reductionism that would

case that even if the exact position and velocities of the particles ofa gas are known it is not possible to integrate all the differentialequations that describe the motions and interactions of theseparticles only the averages magnitudes, as dealt with by theprobabilistic kinetic theory of gases, are considered. Hilbertrecognized the combined use of probability calculations and dif-ferential equations as a very original mathematical contribution,which may lead to deep and interesting consequences (Corry,2004, 169), but warned that it had not been justied as a mathe-matical basis for the KTG and could lead to results that conictedwith the laws of classical mechanics. For example, Hilbert, makinguse of Maxwell's velocity distribution, Boltzmann's logarithmicdenition of entropy and probability theory, pointed out that theresulting law of constant increase in entropy was at odds with theidea of the reversibility of natural phenomena. Next to his recur-rent concern about the reliance on averages,32 another morefundamental probability-related problem of the then current KTGwas that

if Boltzmann proves [that] the Maxwell distribution [is] themost probable one from among all distributions [. . .], thistheorem possesses in itself a certain degree of interest, but itdoes not allow even a minimal inference concerning the velo-city distribution that actually occurs in any given gas.33 [T]heprobability of occurrence of the Maxwell velocity distribution isgreater than that of any other distribution, but equally close tozero, and it is therefore almost absolutely certain that theMaxwell distribution will not occur (Hilbert 1911/1912 quotedin Corry, 2004, 240, my emphasis).34

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 284434form the basis for Hilbert's proposal for a unied foundation ofphysics in 1915. Hilbert remained committed to the general projectof the axiomatization of physics of 1900, but under the inuenceof the increasing mathematical difculty that affected the treat-ment of disciplines based on the atomistic hypothesis, and aboveall the [KTG] (Corry, 1999a, 3),29 Hilbert became convinced thatthis difculty could only be resolved by changing the most foun-dational assumption behind it. This section describes the changesin Hilbert's basic attitude vis--vis the axiomatization of physicsfrom the viewpoint of the mathematical method that partlycaused these difculties, namely probability theory.

3.1. The kinetic theory of gases (KTG), 19111912

After having expressed his support for the atomistic hypothesisas the possible basis for the reduction of the whole of physics tomechanics in his course on mechanics (WS 1910/1911) and con-tinuum mechanics (SS 1911), Hilbert devoted himself to the KTG a theory which itself seemed to give additional support to thefoundational role of mechanics as a unifying, explanatory scheme(Corry, 2004, 46). Hilbert taught a course on the topic in the wintersemester of 1911/1912 and published a paper entitled Begrndungder kinetischen Gastheorie in 1912 (Hilbert, 1912b).30 Both thecourse and the paper31 made explicit the meaning of Hilbert'spurely analytical work on the theory of integral equations (Hilbert,1912a) for the so-called MaxwellBoltzmann equation, or prob-ability distribution function (e.g. Rowlinson, 2005), and empha-sized once more the difcult consequences of the combined use ofdifferential and probability calculus in the KTG (see Corry, 1999a,Section 3; Corry, 2004, 228229 and 237241). In brief, Hilbertsearched for solutions to the MaxwellBoltzmann equation andpursued the question whether the equation can be logicallyderived from the time-reversible equations of classical mechanics.Hilbert's own negative answer gave rise to a problem that occu-pied him in the period 19151923, namely that of determining theobjective meaning of irreversibility (see Section 4.2.1.1.).

The MaxwellBoltzmann distribution, rst described by Max-well in a paper of 1860 and modied and generalized by Boltz-mann in 1877 (Maxwell, 1860; Boltzmann, 1877 [1909]), describesthe probability distribution for the number of molecules havingany given velocity. Maxwell and Boltzmann assumed that in orderto calculate most of the observable properties of a gas it is notnecessary to know the positions and velocities of all particles at agiven time. [I]t sufces to know the average number of moleculeshaving various positions and velocities' (Corry, 2004, 46). Or, inwords drawn from Hilbert's (1905) lecture course, since it is the

28 In brief, the main goal of the contributions of Minkowski was to explorewhether adding the new principle to the already well-established theories andprinciples of mechanics would lead to internal contradictions.

29 Another main factor responsible for the changes in Hilbert's views in thisparticular period was the axiomatic method itself; Hilbert seemed to have alwaysbeen prepared to alter his foundational views, for example on the structure ofmatter, when the axiomatic method impelled him to do so.

30 The content of this paper had been presented to the Gttingen MathematicalSociety in December 1911.

31 This paper rst appeared as Chapter 22 of the book on the theory of integral

equations.What is needed for the theory of gases, Hilbert continued, is aproof of the fact that for a specied distribution, there is a prob-ability very close to 1 that that distribution is asymptoticallyapproached as the number of molecules becomes innitely large(Hilbert 1911/1912 quoted in Corry, 2004, 240).35 Hilbert, therebyloosely referring to the probabilistic idea of convergence to the

32 Corry explains that by 1912, some progress had been made on the solutionof the MaxwellBoltzmann equation. The laws obtained from the partial knowl-edge concerning those solutions, and which described the macroscopic movementand thermal processes in gases, seemed to be qualitative correct. However, themathematical methods used in the derivations seemed inconclusive and some-times arbitrary. It was quite usual to depend on average magnitudes and thus thecalculated values of the coefcients of heat conduction and friction appeared asunreliable' (Corry, 1999a, 8).

33 Here, Hilbert mentioned the following comparative example: In order to laybare the core of this question, I want to recount the following example: In a rafewith one winner out of 1000 tickets, we distribute 998 tickets among 998 personsand the remaining two were give to a single person. This person thus has thegreatest chance to win, compared to all other participants. His probability ofwinning is the greatest, and yet it is highly improbable that he will win. Theprobability of this is close to zero (Um den Kernpunkt der Frage klar zu legen, willich an folgendes Beispiel erinnern: In einer Lotterie mit einem Gewinn und von1000 Loses seien 998 Losen auf 998 Personen verteilt, die zwei brigen Lose mgeeine andere Person erhalten. Dann hat diese Person im Vergleich zu jeder einzelnenandern die grssten Gewinnchancen. Die Wahrscheinlichkeit des Gewinnen ist fursie am grssten, aber es ist immer noch hochst unwahrscheinlich, dass sie gewinnt.Denn die Wahrscheinlichkeit ist so gut wie Null) (Hilbert 1911/1912 quoted inCorry, 2004, 240).

34 Wenn z.B. Boltzmann beweist [. . .] dass die Maxwellsche Verteilung [. . .]unter allen Verteilungen von gegebener Gesamtenergie die wahrscheinlichste ist,so besitzt dieser Satz ja an und fr sich ein gewisses Interesse, aber er gestattetauch nicht der geringsten Schluss auf die Geschwindig-keitsverteilung, welche ineinem bestimmten Gase wirklich eintritt [. . .] [D]ie Wahrscheinlichkeit fr denEintritt der Maxwellschen Geschwindigkeitsverteilung [ist] zwar grsser als die frdas Eintreten einer jeden bestimmten andern, aber doch noch so gut wie Null, undes ist daher fast mit absoluter Gewissheit sicher, dass die Maxwellsche Verteilungnicht eintritt (Hilbert 1911/1912 quoted in Corry, 2004, 240).

35 The passage in which these phrases occur is the following: Was wir fur dieGastheorie brauchen, is sehr viel mehr. Wir wunschen zu beweisen, dass fur einegewisse ausgezeichnete Verteilung eine Wahrscheinlichkeit sehr nahe an 1 besteht,

derart, dass sie sich mit Unendliche wachsende Molkulzahl der 1 asymptotisch

limit (e.g. Fischer, 2011; Hald, 2007), argued that in order to obtainthis proof or, more generally, to provide a justication of prob-ability theory in the KTG, it should become possible to formulatethe question in terms such as these: What is the probability for theoccurrence of a velocity distribution that deviates from Maxwell'sby no more than a given amount? And moreover; what alloweddeviation must we choose in order to obtain the probability 1 inthe limit? (Hilbert 1911/1912 quoted in Corry, 2004, 240).36

Hilbert also discussed additional difculties arising from the

substantially more thorough [treatment] on the basis of theatomic theory (Hilbert 1911/1912 quoted in Schirrmacher, 1999,5, f. 53).38 Hilbert insisted that even though a phenomenologicalapproach la Voigt is indispensable as a makeshift stage on theway to knowledge, it must urgently be left behind in order topenetrate into the real sanctuary of theoretical physics (Hilbert1911/1912 quoted in Schirrmacher, 2003, 10).39

The second, atomistic, approach to physical theories aimed forthis very goal by means of the creation of an axiomatic system thatholds for the whole of physics and which enables all physicalphenomena to be explained from a unied point of view (Hilbert

40

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 2844 35application of probabilistic reasoning in the KTG. Although hesuggested how these were to be mathematically resolved, Hilbertacknowledged that he could not yet give the nal answers. It,indeed, seems to be the case that the more Hilbert becameinvolved with the study of kinetic theory, and at the same timewith the deep mathematical intricacies of the theory of linearintegral equations (Corry, 2004, 267), the more his concerns aboutthe problems implied by the combined use of probability theoryand analysis increased.

3.1.1. The 1911/1912 course on KTG: three approaches to physicaldisciplines

In the introduction to his lecture course on the KTG of thewinter semester of 1911/1912,37 Hilbert distinguished threealternative approaches to physics. The rst was purely phe-nomenological, the second assumed the atomistic hypothesis,and the third aimed at a fundamental (molecular) theory ofmatter.

The rst approach was entertained by, for instance, Hilbert'scolleague, the theoretical physicist Woldemar Voigt (1850-1919).Reecting the Knigsberg tradition (Olesko, 1991, 387388),Voigt upheld the view that experimentally grounded physicaltheories should describe phenomena completely and in eithersimple, direct terms or straightforward equations that directlycorrespond to the empirical features of the phenomena (seeCorry, 2004, 79, 7980, Katzir, 2006, 145147). Given his(mathematical) phenomenology, Voigt, for instance in hisPhnomenologische und atomistische Betrachtungsweise of1915, dismissed atomistic-reductionist explanations andremained working on theories (e.g. piezo-electricity) related tonineteenth-century elds of research such as optics and crys-tallography from a somewhat eclectic non-mathematical, physi-calvisual perspective (Voigt, 1915; see also Katzir, 2003; Jung-nickel & McCommarch, 1986, Chapter 19). Together with theexperimental physicist Eduard Riecke (18451915), Voigt repre-sented the Mathematical Physics Institute at Gttingen. Butbecause of their phenomenological approach to physics bothscientists were isolated not only from Hilbert's circle in specic,but also from the kinds of interest pursued by [the] colleagues inGermany and elsewhere in Europe (Corry, 2004, 234) at large.Hilbert, in his 1911/1912 lecture course on the KTG, criticizedVoigt for fragmenting physics into various chapters each ofwhich is treated using different assumptions, peculiar to each ofthem, and deriving from these assumptions different mathema-tical consequences (Corry, 1999a, 492). He also reproached himfor rejecting the general idea that physics could be given a

(footnote continued)annahert (Hilbert 1911/1912 quoted in Corry, 2004, 240). See also Hilbert's Vortragber meine Gas. Vorlesung (Hilbert Cod. Ms. 588) for this point.

36 [D]ie Frage etwa so zo formulieren: Wie gross ist die Wahrscheinlichkeitdafur, das seine Geschwindigkeitsverteilung eintritt, welche von der Maxwellschennur um hochstens einen bes-timmten Betrag abweicht und weiter: Wie grossmussen wir die zugelassenen Abweichungen wahlen, damit wir im limes dieWahrscheinlichkeit eins erhalten? (Hilbert 1911/1912 quoted in Corry, 2004, 240).

37 This course was announced as a course on continuum mechanics, but itsAusarbeitung, by Hilbert's assistant Erich Hecke (18871947), was entitled Kinetic

theory of gases.1911/1912 quoted in Corry, 2004, 236). Another difference is thatwhere the phenomenological approach solely made use of partialdifferential equations, the atomistic approach employed mathe-matical methods that can be subsumed under the entirely differ-ent and not yet fully developed probability calculus (Schirrma-cher, 2003, 10) as found in the KTG and radiation theory (seeHilbert, 1912a, Chapter 12; Hilbert, 1912b,, 1912c [2009], see alsoCorry, 1998). Because mathematical analysis is not yet developedsufciently to provide for all [its] demands (Corry, 2004, 237) theatomistic approach does not have at its disposal rigorous logicaldeductions and must be satised with rather vague mathematicalformulae (Hilbert 1911/1912 quoted in Corry, 2004, 237).41 At thesame time, the use of probabilistic methods did lead to interestingnew results that seemed to correspond to the experimental facts.

The third approach was to aim for the development of amolecular theory of the structure of matter that is based onatomic theory and would permit all physical properties to bededuced from something even deeper than the system of axiomscalled for in [the atomistic] approach (Schirrmacher, 2003, 11).Hilbert introduced this molecular approach as having to beaccompanied by an advanced or novel mathematics (see Schirr-macher, 2003, 1011) one that would go beyond the atomists'probability theory as far as its degree of mathematical sophisti-cation and exactitude is concerned (Corry, 2004, 237) and thatwould somehow reveal the identity of the (mathematical) modeland (physical) reality. What this third approach should look like interms of its unifying power, knowledge status and mathematicalmethod (see Schirrmacher, 2003, 10) was not made clear in the1911/1912 lecture course, but Hilbert promised to consider indetail the molecular theory in the following year, which he did inseveral courses taught, in 19121913, under the header of 'math-ematical foundations of physics' (see Section 3.2). Both in thesecourses and in subsequent publications and public lectures, Hilbertput forward his theory of linear integral equations as the mathe-matical framework suited to formulate and resolve conceptualgaps (Sauer & Majer, 2009, p. 439) in such elds as kinetic andradiation theory.

Taken together, by 19111912 Hilbert seemed to have envi-sioned his project of the axiomatization of physics from theviewpoint of mechanistic reductionism as follows:

38 [W]esentlicht tiefer eindringend kann man die theoretische Physik aufGrund der Atomtheorie behandeln (Hilbert quoted in Schirrmacher, 2003, 5, f. 53).

39 The full passage goes as follows: Wenn man auf diesem Standpunkt steht, sowird man den frheren nur als einer Notbehlf bezeichnen, der ntig ist als eineerste Stufe der Erkenntnis, uber die man aber eilig hinwegschreiten muss, um in dieeigentlichen Heiligtumer der theoretischen Physik einzudringen (Hilbert 1911/1912 quoted in Schirrmacher, 2003, 10). Schirrmachers translation has here beenslightly amended.

40 The full passage goes as follows: Hier ist das Betreben, ein Axiomensystemzu schaffen, welches fr die ganze Physik gilt, und aus diesem einheitlichenGesichtspunkt alle Erscheinungen zu erklren (Hilbert 1911/1912 quoted in Corry,2004, 236, f. 30).

41 [. . .] sich mit etwas verschwommenen mathematischen Formulierungen

zufrieden geben muss (Hilbert, 1911/1912 quoted in Corry, 2004, 237, f. 32).

After a rst step in understanding by phenomenologicalmeans [and] and [a] successful axiomatization on the basis ofthe atomic theory, [the] objective [. . .] was to relate the basicnotions employed in the axiomatization of physics to actualphysical objects (Schirrmacher, 2003, 11, my emphasis).

Hilbert himself had been aware that the justication for thebelief in the validity of the atomistic hypothesis that, rst impli-citly and later explicitly, accompanied his mechanical reduction-ism was the prospect that it would provide a more accurate anddetailed explanation of natural phenomena once the tools weredeveloped for a comprehensive mathematical treatment of the-ories based on it (Corry, 2004, 267). After the 1911/1912 lecturecourse, his main task became that of addressing the physicaldomain of the molecular theory of the structure of matter itself not in the least because it promised to give a mathematically exact

After his brief consideration of the molecular theory, Hilbertsoon devoted himself to electron theory or, more specically, tothe application of the kinetic theory to the study of the motion ofthe electron. In the summer of 1913 Hilbert organized a series oflectures on the theory and gave a course in which he presented theelectron theory as the foundation of the whole of physics (Hilbert1913 quoted in Corry, 2004, 271).46 The course contained anexplicit treatment of the status of Lorentz covariance and Min-kowski's word-postulate (Welt-postulate)47 as the fundamentalprinciples of the new relativistic physics a topic that Hilbert hadalready briey touched upon in his 1910/1911 course onmechanics. Following Minkowski's macroscopic (or phenomen-ological) approach and Max Born's (18821970) microscopicapproach to establishing the validity of the basic principle ofrelativity (see Corry, 1999b), in his course Hilbert described thegoal of reconstructing the whole of physics in terms of as few basicconcepts as possible as follows:

The most important concepts are the concept of force and ofrigidity.48 From this point of view the electrodynamics wouldappear as the foundation of all of physics. But the attempt todevelop this idea systematically must be postponed for a lateropportunity. In fact, it has to start from the motion of one, of two,etc. electrons, and there are serious difculties on the way to suchan understanding (Hilbert 1913 quoted in Corry, 2004, 272).49

Hilbert explained these difculties as arising from the need tointroduce probabilistic mathematical considerations from kinetic

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 284436description of natural phenomena as they actually occurred inreality, rather than merely an approximation of their possibleoccurrence expressed in terms of averages.

3.2. Theories of the structure of matter and relativity, 19121914

The 1912/1913 lecture course on the molecular theory ofmatter further pursued the idea that the way in which themathematical difculties of the atomistic hypothesis could beresolved would be to adopt a so-called physical point of view.Hilbert suggested to make clear through the use of the axiomaticmethod, those places in which physics intervenes into mathema-tical deduction (Corry, 1999a, 498499) such that it becomespossible to separate three levels of any physical theory:

[First] what is adopted as a logically arbitrary denition ortaken as an assumption of experience, second, what could beconcluded a priori from these assumptions but cannot beconcluded with certainty given current mathematical difcul-ties, and third, what is a proven mathematical conclusion(Hilbert 1912/1913 quoted in Corry, 1999a, 499).42

The course itself also consisted of three parts; where the rstpart discussed certain properties of matter related to the stateequation for a completely homogeneous body understood as amechanical system of molecules, the second part presented morecomplex physical and chemical properties of matter (see Corry,2004, 267270). The third part expressed these results in the formof new axioms.43,44 Because the a priori derivation of these axiomsfrom mechanical principles was to be done in kinetic terms, it was,once again, necessary to have recourse to the fundamental prin-ciple of statistical mechanics,45 namely that the states of a phy-sical system are equally probable.

42 Corry's translation has been slightly amended. The original passage goes asfollows: [W]ir [werden] voneinander trennen, was erstens logisch willkrlicheDenition oder Annahme der Erfahrung entnommen wird, zweitens das, was apriori sich aus diesen Annahmen folgern liesse, aber wegen mathematischerSchwierigkeiten zur Zeit noch nicht sicher gefolgert werden kann, und dritten, das,was bewiesene mathematische Folgerung ist (Hilbert quoted in Corry, 1999a, 499,f. 16).

43 Hilbert wrote that Um in einzelnen Falle die karakteristische Funktion inihrere Abhngigkeit von der eigentlichen Vernderlichen und den Massen derunabhngigen Bestandteile zu ermitteln, mssen verschiedenen neue Axiomehinzugezogen werden (Hilbert, 1912/1913 quoted in Corry, 2004, 269, f. 136).

44 It is important to note, with Corry, that Hilbert never performed for aphysical theory exactly the same kind of axiomatic analysis he had done for geo-metry, though he very often declared this to be the case. Also, his derivations of thebasic laws of the various disciplines from the axioms were always rather sketchy,when they appeared at all (Corry, 1999d, 15).

45 The full passage goes as follows: Um die empirisch gegebenen und zumathematischen Formeln verallgemeinerten Ergebnisse [. . .] a priori und zwar auf

rein mechanischemWegen abzuleiten, greifen wir wieder auf des Grundprinzip destheory as soon as the description concerns the motion of andinteractions between more than a single electron (the so-called n-electron problem).50 In other words, the situation was such thatbecause the explanation of the magnetic and electric interactionsamong electrons in mechanical terms is only an approximation itmust be admitted that we [either] only speak [. . .] of averages(Corry, 1999a, 500) or settle for describing the motion of oneelectron. Although he seems to have been more aware than ever ofthe mathematical and physical difculties [. . .] associated with aconception of nature based on the model underlying kinetic the-ory (Corry, 2004, 273) and even proposed, in one speciccontext,51 to substitute the mechanical approach derived from the

(footnote continued)statischen Mechanik zurck, von der wir bereits im ersten Teil ausgegangen waren(Hilbert 1912/1913 quoted in Corry, 2004, 270, f. 141).

46 Die Elektronentheorie wrde daher von diesem Gesichtpunkt aus dasFundament der gesamten Physik sein (Hilbert, 1913 quoted in Corry, 2004, 271,f. 150).

47 It was Minkowski who rst referred to the principle of covariance or rela-tivity as the word-postulate in his Space and Time of 1909. Here, he wrote thatthe postulate comes to mean that only the four-dimensional world of space andtime is given by phenomena, but that the projection in space and in time may stillbe undertaken with a certain degree of freedom, I prefer to call it the postulate ofthe absolute world (or briey the world-postulate) (Minkowski, 1909 [1952], 104).

48 At this point, it is interesting to observe, with Renn and Stachel, that [i]nspite of the conceptual revolution brought about by special relativity concerningthe revision of the concepts of space and time but also the conceptual autonomy ofthe eld concept from that of the ether, Hilbert nevertheless continued to count ontraditional concepts such as force and rigidity as the building blocks for his axio-matization program (Renn & Stachel, 1999, 78).

49 Die wichtigsten Begriffe sind die der Kraft und der Starrheit. Die Elek-tronentheorie wrde daher von diesem Gesichtspunkt aus das Fundament dergesamten Physik sein. Den Versuch ihres systematischen Aufbaues verschieben wirjedoch auf spter; er htte von der Bewegung eines, zweier Elektronen u.s.w.auszugehen, und ihm stellen sich bedeutende Schwierigkeiten in der Weg, daschon die entsprechenden Probleme der Newtonschen Mechanik fr mehr als zweiKorper ungelst sind (Hilbert 1913 quoted in Corry, 2004, 272, f. 152).

50 There was, of course, also a highly complex system of integro-differentialequations involved in the search for the equations of motion for a system ofelectrons.

51 Corry (2004, 272) refers to the fact that Hilbert, in order to describe theconduction of electricity in metals, [. . .] developed a mechanical picture derivedfrom the theory of gases, which he then later wanted to substitute by an electro-

dynamical one. The original passage in which this idea appears is the following:

theory of gases by an electromagnetical one, Hilbert remainedcommitted to the atomistic-mechanical outlook. The 1913 lecturecourse on the molecular theory of matter even contained thestatement that atomism is to be considered as a necessary con-sequence of the principle of relativity.52

It was in his lecture course on electromagnetic oscillations ofthe winter semester of 1913/1914 that Hilbert made explicit for therst time his view that the whole of physics arises from electro-dynamics, rather than from mechanics53 and that relativisticmechanics, or four-dimensional electrodynamics was on theverge of being assimilated by mathematics (Hilbert 1913/1914quoted in Corry, 2004, 280).54 However, Hilbert also acknowledgednot only that [w]e are really still very distant from a full realiza-tion of [. . .] reducing all physical phenomena to the n-electronproblem (Hilbert 1913/1914 quoted in Corry, 2004, 281),55 but alsothat if one can do little with the n-body problem, it is even lessfruitful to proceed on the basis of the treatment of the n-electronproblem.56 These difculties of the reductionist program forcedHilbert to make two concessions. Firstly, instead of a mathema-tical foundation based on the equations of motion of the electrons,we still need to adopt partly arbitrary assumptions, partly tem-porary hypothesis [and] certain very fundamental assumptionsthat we later need to modify (Hilbert 1913/1914 quoted in Corry,2004, 281).57 Hilbert did not describe these three kinds ofassumptions, but it seems plausible that he referred to the ato-mistic hypothesis and the earlier mentioned condition of equi-probability. Secondly, Hilbert had it that with regard to the n-electron problem and thus, more generally, to the further devel-opment of electrodynamics, the point for us is rather to silence[verstmmeln] the [problem], [to] integrate the simplied equa-tions and to ascend from their solutions to more general solutions

Where the phenomenological approach and its partial differ-ential equations were merely a rst step in the understanding ofnatural phenomena, the foundational problems that accompaniedthe atomistic approach and its probabilistic methods requiredfurther investigation into the (molecular and electron) theory ofthe structure of matter. Hilbert hoped that the novel non-probabilistic mathematics of this theory-of-matter-approachwould reveal the identity of the axiomatic model and physicalreality (Schirrmacher, 2003, 11). But the more Hilbert tried tocome to terms with the fundamental n-electron problem, the morehe seems to have despaired at the insurmountable role of prob-ability theory, as a kind of mathematics rather than a physicaldiscipline, and, more in specic, of the probabilistic method ofmean values.

4. Third period. Probability as an anthropomorphic accessor-ial principle of human thought in Hilberts epistemologicalreections on the Foundations of Physics: 19151923

The pre-history of Hilbert's two notes entitled the Foundationsof Physics of 191559 and 191760 in which he himself thought tohave accomplished the formulation of the physics in general,rather than just of a particular kind of phenomena (Corry, 2004,33) is marked by two central events. Firstly, the belated61 adop-tion of Born's reformulation, in 1912 and 1913 (see Corry, 2004,309315), of Gustav Mie's (18681957) electromagnetic theory ofthe structure of matter in which the existence of electrons and,

semester (Electron theory) in spite of their obvious, direct connection [. . .] showno evidence of a sudden interest in Mie's theory or in the point of view developed

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 2844 37by means of corrections (Hilbert 1913/1914 quoted in Corry, 2004,281, f. 170).58

(footnote continued)Unser nchstes Ziel ist, eine Erklrung der Elektrizitatsleitung in Metallen zugewinnen. Zu diesem Zwecke machen wir uns von der Elektronen zunchst fol-gendes der Gastheorie entnommene mechanische Bild, das wir spter durch einelektrodynamosches ersetzen werden (Hilbert, 1913 quoted in Corry, 2004, 272,f. 151).

52 Es sind somit die zum Aufbau der Physik unentbehrlichen starren Krpernur in den kleinsten Teilen mglich; man knnte sagen: das Relativittsprinzipergibt also als notwendige Folge die Atomistik (Hilbert, 1913 quoted in Corry, 2004,274, f. 157).

53 Es scheint indessen, als ob die theoretische Physik schliesslich ganz und garin der Elektrodynamik aufgeht, insofern jede einzele noch so spezielle Frage inletzter Instanz an die Elektrodynamik appellieren muss (Hilbert, 1913/1914 quotedin Corry, 2004, 280, f. 169).

54 Nun glaube ich aber, dass es der hchste Ruhm einer jeden Wissenschaft ist,von der Mathematik assimiliert zu warden, und dass auch die theoretische Physikjetzt im Begriff steht, sich diesen Ruhm zu erwerben. In erster Linie gilt dies vonder Relativittsmechanik oder vierdimensionalen Elektrodynamik, (Hilbert 1913/1914 quoted in Corry, 2004, 280, f. 168). It may here be remarked that Hilbert alsopermitted himself to claim that one could [now] divide mathematics [into] one-dimensional mathematics, i.e. arithmetic, then function theory, which essentiallylimits itself to two dimensions; then geometry, and nally four-dimensionalmechanics (Hilbert 1913/1914 quoted in Corry, 2004, 280) ([M]an knnte [. . .]die Mathematik einteilen in die eindimensionale Mathematik, die Arithmetik,ferner in die Funktionentheorie, die sich im wesentlichen auf zwei Dimensionenbeschrankt, in die Geometrie, und schliesslich in die vierdimensionale Mechanik(Hilbert 1913/1914 quoted in Corry, 2004, 280, f. 169)).

55 Von der Verwirklichung unseres leitenden Gedankens, alle physikalischenVorgnge auf das n-Elektronenproblem zurckzufhren, sind wir freilich noch sehrweit entfernt (Hilbert 1913/1914 quoted in Corry, 2004, 281, f. 171).

56 So wenig man schon mit dem n-Korperproblem arbeiten kann, so ware esnoch fruchtloser, auf die Behandlung des n-Elektronenproblemes einzugehen(Hilbert 1913/1914, quoted in Corry, 2004, 281, f. 170).

57 An Stelle einer mathematische Begrndung aus den Bewegungsgleichungender Elektronen mssen vielmehr noch teils willkrliche Annahmen treten, teilsvorluge Hypothesen, die Spter einmal begrndet warden drften, teils aberauch Annahmen ganz prinzipieller Natur, die sicher spter modiziert werden

mssen (Hilbert 1913/1914 quoted in Corry, 2004, 281, f. 171).in it [. . .] Possible, this was connected to the fact that Mie's strong electromagneticreductionism was contrary to Hilbert's current views, which also favored reduc-tionism, but still from a mechanistic perspective at the time (Corry, 2004, 310). Seealso Renn and Stachel (1999, 8).

62 Reecting on Born's reformulation of Mie's theory, Corry (2004, 312) writesthat [w]hereas Lorentzs theory of the electron was based on certain hypothesesconcerning the nature of matter (e.g. the rigidity of the electron) [. . .] Mieattempted to derive mathematically the existence of electrons [. . .] from a modied[i.e. non-linear] version of the Maxwell equations, i.e. without starting from anyparticular conception concerning the nature of physical phenomena.

63 Much has been written on the priority of the discovery of the eld equationsof general relativity. Where the majority of textbooks spoke of Einstein's equa-tions, according to the commonly accepted academic view Hilbert completed thegeneral theory of relativity some ve days before Einstein. Corry, Renn & Stachel(1997) have shown that Hilbert did not anticipate Einstein because the rst set ofthe proofs of his 1915 paper is not generally covariant and does not include theexplicit form of the eld equations of general relativity. In this context see, forinstance, also Earman & Glymour (1978), Lehner, Renn & Schemmel (2012), Mehratherefore, of atoms and matter in general, is mathematically62

derived from the electric eld (e.g. Mie, 1912a,, 1912b; see alsoBorn, 1914; Corry, 1999a,, 1999b,, 2004, 298306; Mehra, 1973,Section 3.4; Renn & Stachel, 1999, 810; Smeenk & Martin, 2007).Secondly, the negotiation of Albert Einstein's (18791955) gen-eralization of the principle of relativity and his theory of gravita-tion of 19131914 (e.g. Einstein & Grossmann, 1913; Einstein, 1914,see also Lehner, 2005; Renn, 2005; Stachel, 1989).63 Hilbert's

58 Es handelt sich vielmehr fr uns darum, das n-Elektronenproblem zu ver-stmmeln, die vereinfachte Gleichungen zu integrieren und von ihren Lsungendurch Korrekturen zu allgemeineren Lsungen aufzusteigen (Hilbert 1913/1914quoted in Corry, 2004, 281, f. 170).

59 The First Communication was originally delivered (as a talk entitled Thefundamental equations of physics) to the Gttingen Academy of Science inNovember 1915, submitted to its Proceedings on 19 November of that year andpublished in March 1916 (Hilbert, 1916 [2009]).

60 The Second Communication underwent several major revisions beforebeing submitted to the Gttingen Academy of Science on 23 December 1916 andappearing in 1917 (Hilbert, 1917a [2009]).

61 As Corry has noted: [T]he lecture notes of the courses Hilbert taught in thewinter semester of 1912/1912 (Molecular theory of matter) and in the following(1973, Chapter 7) and Stachel (1999).

attempt, in the rst note (Hilbert, 1916 [2009]), at a unied eldtheory of electromagnetism and gravitation was a result of acomplex axiomatic synthesis of the speculative physical theoriesof Mie and Einstein (see Renn & Stachel, 1999; Sauer, 1999). Inbrief, the theory consisted of a generally covariant reformulation ofMie's theory of matter and Einstein's theory of gravitation suchthat both could be derived from a single variational principle for aLagrangian and the (number of) eld equations following from thevariational principle would allow for the claim that electro-magnetism is an effect of gravitation. Hilbert originally wanted hissecond note (Hilbert, 1917a [2009]) to be about the physical con-sequences of his own unied eld theory of the rst note, but,much in line with the content of his courses of 1916191764, thepublished version solely concerned the general theory of relativity.Because those lectures, given between the years 1919 and 1923, inwhich Hilbert would return to the status of probability (Section3.2) were written as philosophical reections on issues related toone of the central topics with which he was occupied in 19161917, namely the causality quandary,65 the (background of the)second note is taken up in the following (sub)section(s).

4.1. The causality quandary

Hilberts rst note on the Foundations of Physics of March1916 contained two fundamental axioms: Axiom I: Mie's axiom ofthe world-function66 and II. Axiom of general covariance.67 Theso-called Leitmotiv68 (Theorem I) of the theory stated that in so

theory of differential equations, the requirement of four furthernon-invariant equations to supplement [the gravitational andelectromagnetic] equations is unavoidable (Hilbert, 1916[2009] quoted in Brading & Ryckman, 2007, 73).

Although the published version of the rst note assumed thepossibility of using generally-covariant eld equations not sup-plemented by coordinate restrictions in spite of Einstein's hole-argument71 (Howard, 1992; Howard & Norton, 1993; Iftime &Stachel, 2006),72 in this passage from the December Proofs Hilbertemphasized the necessity of introducing four additional non-covariant equations (or differential relations) in order to obtain aunique determination of the evolution of all 14 variables. Axiom III('The Axiom of Space and Time'), that contained these four equa-

L.M. Verburgt / Studies in History and Philosophy of Modern Physics 53 (2016) 284438far as it is the case that in the system of n differential equations onn variables [. . .] four of these equations are always a consequenceof the other n 469 the equations for the four electromagneticpotentials (gs) are the consequence of the equations for the 10gravitational potentials (g). Because Theorem I shows thatAxioms I and II can only provide ten essentially independentequations for the 14 potentials,70 Hilbert argued that

in order to keep the deterministic character of the funda-mental equations of physics, in correspondence with Cauchy's

64 Hilbert taught a lecture course entitled Foundations of Physics I (generalrelativity) in the summer semester of 1916 and one entitled Foundations of Phy-sics II (general relativity) in the winter semester of 1916/1917 (see Corry,2004, 451).

65 Hilbert delivered a lecture with this title (The principle of causality in 1917(Hilbert, 1917b [2009]).

66 The rst axiom consisted of a variational argument for a scalar Hamiltonian(H) (or Lagrangian (L)) (world)function H with space-time coordinates (orworld-parameters) i.e. (Einstein's) 10 gravitational potentials with their rst andsecond derivatives and (Mie's) four electromagnetic potentials with their rstderivatives determining the coordinates of a four-dimensional manifold. Hilbertused the Hamiltonian to derive the basic equations of the theory, starting from theassumption that, under innitesimal variations of its parameters, the variation ofthe integral

RH

gd

p[. . .] vanishes for any of the potentials (Corry, 1999a, 518).

67 The second axiom postulated that the world-function H (see footnote 63) isgenerally covariant i.e. retains its form with respect to arbitrary coordinate (orworld-parameter) transformations.

68 In brief, Hilbert described this theorem as follows: [I]n the system of ndifferential equations on n variables [. . .] four of these equations are always aconsequence of the other n 4, in the sense that four linearly independent com-binations of the n differential equations and their total derivatives are alwaysidentically satised (Hilbert, 1916 [2009], 31).

69 Hilbert formulated the theorem as follows: Theorem I. Let J be a scalarexpression of n magnitudes and their derivatives that is invariant under arbitrarytransformations of the four world-parameters, and let the Lagrange variationalequations corresponding to the n magnitudes be derived from the integralRJ

g

pd 0. Then, in the system of n differential equations on n variables

obtained in this way, four of these equations are always a consequence of the othern 4, in the sense that four linearly independent combinations of the n differentialequations and their total derivatives identically satised (Hilbert, 1916 [2009], 3031, original translation).

70 Here, Hilbert, 1916 refers to the December Proofs of Hilbert's rst note that

have been preserved in Hilbert's Nachlass (DHN 634).tions, thus allowed Hilbert to extract a physically acceptableCauchy-determinate structure within an otherwise generallycovariant theory (Brading & Ryckman, 2012, 188).

After the appearance of the published version of the rst note,Hilbert repeatedly made it clear that he was still in quandaryabout how to treat the causality issue (Renn & Stachel, 1999, 73).For example, in the lecture course The Foundations of Physics, I ofthe summer semester of 1916, Hilbert noted that the status of thecausality principle (Kausalittsprinzip) within generally-covariantphysics was not yet claried73 (Hilbert, 1916 quoted in Renn &Stachel, 1999, 7) by his own unied eld theory. The newly foundsolution to the problem of causality that Hilbert brought to thefore in his second note on the Foundations of Physics of 1917 hadalready appeared in his undated Causality lecture of 1917 (Hilbert,1917a [2009],, 1917b [2009]). Remarkably, the solution was for-mulated in terms of a revision of Kantian epistemology in light ofthe principle of objectivity implicit in Einstein's general relativity(see Einstein, 1916; Brading & Ryckman, 2012, Section 8.6; Hallet,1994; Majer, 1993; Majer & Sauer, 2005,, 2006; Ryckman, 2008).

4.1.1. The new solution of the Causality lecture and the second noteon the Foundations of Physics

Both the Causality lectures and the second note opened withthe explicit statement that causality cannot be restored on thebasis of the (mathematically false)74 idea of postulating fouradditional non-covariant equations (see Hilbert, 1917a [2009],,1917b [2009], see also Renn & Stachel, 1999, 7475). The startingpoint of the new solution was the distinction between two parts ofthe causality problem; the issue of, rstly, causal ordering (I-CO)and, secondly of univocal determination (I-UD). Where the I-COpertained to the fact that a conict with the [experienced] causalorder [arises] if two world-points [Weltpunkte] lying along thesame time-like curve [Zeitlinie], and standing in [a] relation ofcause and effect, can be transformed so that they become simul-taneous (Brading & Ryckman, 2007, 33), the I-UD amounted to thefact that from the knowledge of physical magnitudes in the

71 After having realized that both the eld equations as well as the law ofenergymomentum conservation of the Entwurf theory of 1913 were not generallycovariant, Einstein devised a highly complex argument (the hole argument) whichwas to prove that there cannot exist generally covariant eld equations thatcompletely determine the eld.

72 It may here be remarked that Einstein's nal presentation of his theory ofgravitation proved that generally covariant eld equations do not need to besupplemented