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    Historia Mathematics 9 (1982) 413-440

    HERBART'S INFLUENCE ON BERNHARDRIEMANNBY ERHARD SCHOLZGESAMTHOCHSCHULE WUPPERTAL,FACHBEREICH 7, GAUSSSTRASSE 20,56 WUPPERTAL 1, FEDERAL REPUBLIC OF GERMANY

    SUMMARIESB. Riemann (1826-1866) knew a great deal of the

    thought of the German philosopher J. Fr. Herbart (1776-1841). During his studies of the philosopher's workhe copied out numerous excerpts and made a few noteswhich are preserved (at least partially) in theRiemann Archiv at Gattingen. This material revealsthat Herbart influenced Riemann much more in hisepistemology and the comprehension of science than inhis particular philosophy of space and spatial think-ing. Thus the relationship between Herbart and Riemannhas to be looked upon as an example of an influence ofGerman Bildungsphilosophie on the mathematics of the19th century.

    Bekanntlich befasste sich B. Riemann (1826-1866)griindlich mit der Philosophie J. Fr. Herbarts (1776-1841). Er fertigte wahrend seiner Herbart-Studienumfangreiche Exzerpte und einige Notizen an, die(zumindest teilweise) im Gijttinger Riemann Archiverhalten sind. Dem Inhalt dieser Materialien nach zuschlieben lag der Schwerpunkt des HerbartschenEinflusses auf Riemann nicht so sehr in seiner Philo-Sophie des Raumes und des r&mlichen Denkens alsvielmehr im Bereich der Erkenntnistheorie und Wissen-schaftsauffassung. Gerade darin stellt sich aber dieBeziehung zwischen Herbart und Riemann als Fallbeispieleines Einflusses der deutschen Bildungsphilosophie aufdie Mathematik des 19. Jahrhunderts dar.

    Bernhard Riemann was one of the most inf luential mathema-ticians of the last century, working in mathematical fields asdifferent as complex function theory, partial differential equa-tions, foundations of Fourier analysis, topology, differentialgeometry, birational geometry, and number theory [Dieudonne

    0315-0860/82/040413-28$02.00/OCopyright 0 1982 by Academic Press, Inc.All rights of reproduction in any form reserved.413

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    1978, Klein 1926, Kline 19721. His work, although small involume, had a strong impact on later researchers. Because ofits deep conceptual penetration of the subject matter in question,it became a source of inspiration when the tendency of mathema-tics toward a clarification of the underlying main concepts be-came stronger in the second half of the 19th century [l]. Thusit is of interest to investigate the background of Riemann'sresearch related to problems of conceptual structure.Riemann's published works contain philosophical fragments[1892a, 509 ff.] which shed some light upon his reflectionsabout science. They also provide evidence that Riemann wasstrongly influenced by the philosopher Johann Friedrich Herbart,who had been teaching at Kdnigsberg and Gdttingen until hisdeath in 1841 (see Appendix la). Riemann stated:

    The author [Riemann himself] is a Herbartian inpsychology and epistemology (methodology and eidolology);in most cases, however, he cannot agree with Herbart'snatural philosophy and the metaphysical disciplines(ontology and synechology) referring to it. [1892a, 5081This statement has been cited often, and so some influenceof Herbart on Riemann is generally assumed in the literature[Russell 1956, 62 f.; Torretti 1978, 107 f.; Scholz 1980, 94-951.But there have been differences of opinion over which of Herbart'sideas were most inf luential. Russell believed that it wasHerbart's philosophy (and psychology) of space which made a deepimpact on Riemann. That would not correspond to the contents ofthe citation above, however, because "synechology" [Synechologie]was Herbart's name for the philosophical foundation of the con-cept of space, and this was exactly one of the points Riemannsaid had failed to convince him. But this alone will not justifyrejecting Russell's statement without further consideration, forit gains some plausibil ity from a similarity between Riemann'sexplanation of the concept of manifold and Herbart's "serialforms" (see below). Torretti's discussion of the topic agreesessentially with that of Russell, although he does not find allof it acceptable. In (implicit) contrast to this, Schnlz [1980]looks for the link between Herbart and Riemann much more on thegeneral level of epistemology, which, again, played a distinctrole in Riemann's formation of the concept of manifold.But al l of these estimations suffer from the limitation ofthe material on which they are founded. On the basis of thepublished works of Riemann and Herbart one couldrdo no more thantry to make reasonable guesses. But the Riemann Nachlass atGijt tingen mniversity library contains excerpts and notes fromRiemann's studies of Herbart, which give a clearer picture thanbefore of Herbart's philosophy and how it was of particular

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    interest to Riemann. This article presents this previouslyunpublished material, discusses the most essential points ofHerbart's philosophy to which Riemann responded, and finallyoffers a short evaluation which might point to questions forfurther investigations.

    THE MATERIALFolder 18 of Riemann's Nachlass contains 203 sheets withmanuscripts classified as "Fragmente naturphilosophischenInhalts"; but there are at least 12 sheets with notes and ex-cerpts from Riemann's studies of Herbart (numbers 57, 59, 60,64, 66, 70, 72, 73, 90, 140, 141, 171) [2]. The excerpts aretaken from Herbart's works on metaphysics and psychology[Herbart 1796, 1799, 1811, 1812, 1822, 1824, 1825, 18281. Inaddition, there is a note (R.177) [3] with references to[Herbart 18071 and another (R.141) with references to [Herbart18511.Herbart distinguished two large fields of philosophy, meta-

    physics and aesthetics and practical philosophy. Metaphysics,according to Herbart, contained four disciplines: Methodology,eidolology [Eidolologie], ontology, and synechology [Synecho-logic] .Trained in the philosophy of Kant and Fichte, he saw thegoal of philosophy and the sciences as advancing from contra-dictory sense perceptions to concepts of the underlying reality.This advancement he believed should be governed by sound princi-ples of methodology. His thorough studies of Fichte's philos-ophy of the self led Herbart to the insight that the self wasone of the most subtle concepts of philosophy, endowed with amultitude of contradictions (see Appendix la). To show this,and to come to grips with it, was the task of the disciplinehe called eidolology. Methodology and eidolology together canbe looked upon (as did Riemann in the citation above) as formingHerbart's epistemology. The most general categories of reality(for Herbart: being [Sein], quality, inherence, and change)formed the subject matter of his ontology. However, this wasnot yet sufficient as a foundation for scientific investigationand had to be completed by generation of the concepts of space,time, number, and matter, which was done in the discipline ofsynechology [Weiss 19281.Thematically, al l four disciplines of Herbart's metaphysicsare covered by Riemann's excerpts, whereas aesthetics and prac-tical philosophy are touched upon only marginally (e.g., R.60r).Table 1 provides a list of the most important of these excerptswith reference to topic, source (in Herbart's Werke) , disciplinein Herbart's classification of metaphysics, and folio number

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    TABLE 1

    Survey of Riemann's excerpts from Herbart in folder 18of Riemann's Nachlass, HandschriftenabteilungUniversit;itsbibliothek Gijttingen

    Subject matter Excerpt from FolioHerbart (fol.der 18)

    (i) MethodologyPhilosophical concepts aschanging products of thoughtReality lying behind per-ceptionsReality lying behind per-ceptionsMethod of relations

    (ii) EidolologySchelling's dialectics ofthe selfHerbart's explanation ofthe selfHerbart's criticism ofFichte's self(iii) Ontology

    Things as bundles ofpropertiesSubstance as understood byLeibniz, Locke, and KantFarreaching character ofchange of chemical sub-stancesDifficulties of Greek phi-losophers with contradic-tions of changeCausalityCausality

    (iv) SynechologySerial formsSerial formsObjective basis of spaceimagination

    [1825, 198/199][1825, 199/200][1828, 123/124][1851, 594-5961

    11796, 30/31] 64r[1799, 108-1101 64v[1822, 107/108] 64v

    [1825, 193111825, 193/l.94, 1963

    [1828, 130/131][1828, 134,'135, 330-334, 337, 3401[1825, 115, 424/425][1828, 1221

    [1825, 192/193][1824, 428, 4211[1824, 4251Criticism of Kant's idea ofspace [1824, 4281

    5gr5gv

    66r

    5gr72', 73v7ov72+

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    HM9 Herbart's Influence on Bernhard Riemann 417

    in Riemann's Nachlass, folder 18. Apart from the excerptsthere are three notes by Riemann referring to Herbart (R-140,141, 177). They are documented in Appendix 2 and note 17.An analysis of the excerpt material and the addit ional notesshows that there are certain points in Herbart's philosophywhich seemed of particular importance to Riemann: elements ofdialectics, methodology, spatial concepts, and the orientationof mathematical research. These points are discussed in thefollowing sections, in order to draw a picture of Herbart'sinfluence on Riemann.

    ELEMENTS OF DIALECTICSPhilosophy of the Self (eidolology)

    Riemann came into contact with Fichte's and Schelling'sphilosophy of the self by way of the early works of Herbart[1796, 1799, R.641. At the beginnings of his studies (1794),Herbart had been one of Fichte's main disciples and was fasci-nated by the idea of the "absolute self" as the basis for cog-nition of al l of reality [Asmus 1968, 98 ff.]. But alreadyduring the final phase of his studies (1795/1796) he developed--in his criticism of Fichte's idea--the foundations of his ownphilosophy of the self, which he later called eidolology.Herbart's argument was based on the observation that theconception of the self always contained in itself the contra-diction between the act of reflection and the reflected, andso in Fichte's terminology of the self and the not-self [Ichund Nicht-Ich]. This contradiction, however, served him asproof that the self was no simple idea; on the contrary, it wasone of the richest and most concrete concepts, and so could notbe the starting point for philosophy. This led him to regardFichte's "absolute self" as an idealist fiction [Herbart 1822,107-108*; 1824, 237 ff.] [41. The self seemed to him to be abundle of properties [Complexion von Merkmalen] [1828, 209/210]and, according to Herbart, it contained a fluctuating structureof ideas or presentations [Vorstellungen] opposed to each otherand impeding one another [1799, 108 f.*]. This appeared to himto be important because he regarded the process of gainingknowledge as governed not only by the principles of methodologybut also by the laws of the movement and impediments of presen-tations.Riemann excerpted the essential points of Herbart's argu-mentation, was obviously convinced by it, and adopted the basicfeatures of the philosopher's eidolology/psychology. This, al-ready stated in the citation above, is made completely clear bya note in which Riemann summarized Herbart's position in his

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    own words, expressing what he felt were its most importantaspects (R.140; see Appendix 2):

    1. Presentations are the elementary forces of the soul.They oppose and impede each other. The resulting antagonismsgenerate the movement o f thought.

    2. Images of the outside world are generated by negationsof the given perceptions. Negation itself has to be learnedfrom the experience of cancellation of presentations.3. Negation is canceled and a new position gained, when a

    connection with the already formed and interdependent images/concepts is attained.Riemann thus came to know and accept certain elements ofHerbart's dialectics in his reception of such eidolology: op-position of concepts/presentations, negation, and position as

    canceled negation.In his published philosophical fragments we find a passagein which he experimented with a conceptual structure of thesis/antithesis [1892a, 518 ff. ]. Previously it was unclear whathis background was for expressing this idea. Now I think ananswer can be given: Riemann's knowledge of certain aspects ofFichte's and Schelling's dialectical philosophy came through hisstudies of Herbart, although Riemann was not directly influencedby their philosophy. Nevertheless, he adopted basic featuresof a dialectical character from Herbart's philosophy, fromeidolology in particular. It was German philosophy of the firsthalf of the 19th century, which Riemann encountered by way ofhis Herbart studies, that set the background for his own dialec-tical arguments.

    The Problem of Change and the Structure of the RealRiemann studied Herbart's historical treatise on metaphysics

    from the Greeks to Schelling and Fries [Herbart 18281 with in-terest and made long excerpts from this work (R.64, 66), includ-ing a long passage about Plato (R.66). Herbart explained thatPlato, once he stumbled upon the contradiction lying in changeas a form of experience, made a strange [ungereimt] distinctionbetween the world o f knowledge containing eternal ideas and theworld of opinions dealing with changing phenomena.

    According to Herbart, Plato tried to mitigate the strange-ness of the sharp opposition of these two worlds by assuminga substance without any qualities, which, shaped by ideas, wouldgenerate the changing things of the sensual world (Herbart 1828,330 f. *] . Herbart, and with him Riemann, traced a line of fol-lowers who somehow adopted this conception, although with modi-ficatons, which ran from Aristotle [Herbart 1828, 134 f. *] tothe scholastics [1828, 323 f. *] , Spinoza, and even to Schelling[1828, 135*].

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    In contrast to these philosophers Herbart saw a much closerconnection between changing phenomena and the real. He was surethat criticism of the phenomena could lead to knowledge about"the real" [das Reale] [5], and he set himself the task ofstructuring that knowledge according to the connections of thephenomena. This was the responsibility of metaphysics as Herbertunderstood it, but metaphysics had to do more. Once it had pen-etrated into the depths of "the real," it had to rise again, inorder to distinguish "the given" (the phenomena) from "the real"as far as possible. This movement of thought, compared by himto an arc, ought to be governed by the principles of methodology[1829, 141.Riemann was obviously interested in this type of argument.He copied out the following explanations of Herbart point bypoint:In the first place, according to Herbart, experience showsus properties and bundles [Complexionen] of properties, theunderlying reality of which must first be sought in things towhich the properties are ascribed [1825, 199*]. But the nextstep of the analysis, so he continued, led investigators to the

    elements of Greek philosophy, from which the things now obtaineda "borrowed reality" [geliehene Realitat, 1825, 1991. Laterthe scientists' analysis led to the introduction of chemicalelements from which now the old elements (at least water andair, which Herbart explicit ly referred to) were derived. Finallyidealist philosophy came, according to Herbart, and reduced evenchemical elements, as well as properties, things, and old ele-ments, to intuition [Anschauung] and thought, concluding thatthe idea of the self underlay al l other concepts and "lent real-ity" to them [Herbart 1825, 2001 [6].We see from this argument that Herbart did not restrict hisreflections to metaphysics proper but also included science.This goes hand in hand with his opinion concerning the relation-ship between philosophy and science, which rejected a sharpdividing line between the two.From Riemann's selection of passages we may conclude thathe was particularly interested in the question of how the problemof change and its consequences for the structure of reality hadbeen tackled in metaphysics [7]. He noted Herbart's conceptionsof things as bundles [Complexionen] of properties (R.59), behindwhich lay the real [Herbart 1828, 199-2001. But the specificconstruction of "the real" as proposed by Herbart is not reflecedin Riemann excerpts. This corresponds perfectly well with hisdeclaration that he did not agree with the philosopher's ontol-

    WY - The distinction, however, between the phenomena and amore stable underlying reality with an intense relationship be-tween both became an essential point in Riemann's own reflectionsabout the epistemology of science [1892b].

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    THE PROCESS OF KNOWLEDGE (METHODOLOGY)

    Riemann adopted Herbart's idea of the two-phase movement("arc") for the advancement of knowledge (see above), althoughhe preferred a terminology which was closer to that of scientific

    research than to that of metaphysics. So in a note which can belooked upon as an intermediate paper coming somewhere betweenthe pure Herbart study and his own epistemological essay [1892b],he characterized these two phases in the following way:

    1. Formation of concepts from perceptions--abstraction and induction, synthesis a posteriori2. Generation of the perceived from the concepts--synthesis a priori. [R.l41; see Appendix 21Obviously Riemann did not mean what Kant did by synthesis apriori. He had a relative a priori in mind, which, serving as

    a conceptual framework for experience, presupposed the otherhalf of the whole process of knowledge (abstraction and induc-tion). Thus the historical change of concepts was not onlypossible but was to be achieved as follows:

    3. Change (or completion) of the concepts as small aspossible, where the perceived is impossible orunlikely according to the concepts. [R. 1413It seemed to him that Herbart's "method of relations" which

    dealt with the dialectics of reason and consequence (see Appendixlb) was useful for accomplishing this change.This was not the only reference to Herbart, showing thatRiemann's ideas about change of concepts were partially due tothe philosopher. He also referred to Sections 139-145 of Herbart's

    "Science of Psychology" [1825] from which he made long excerpts(R.59). There he noted in particular that Herbart looked uponchanges of ideas about reality resulting from the solution ofcontradictions as conceptual revolutions similar to those inastronomical knowledge [Herbart 1825, 198*], thus consideringthe process of knowledge as a stepwise acquisition of levels inthe intellectual evolution of man [Bildungsstufen . . . . welchesuccessiv erreicht werden, Herbart 1825, 199*] [8].

    It seems clear, then, that two important features of Riemann'sdevelopmental dialectics as proposed in his published epistemo-logical fragment [1892b] were due to Herbart:--a distinction between the phenomena and the underlying

    reality with a corresponding difference between the perception

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    and the conceptual acquisition of reality; the latter had "togo back behind the phenomena" and according to Riemann couldthus serve as a base for an explanation of the former;--a belief in progress of knowledge by "transformation ofolder conceptual systems" because of contradictions on the levelof the explanations or between concepts and phenomena.But Riemann's presentation differed in three points fromthat of Herbart [Scholz 1980, 94-961:--Riemann referred mainly to the sciences, whereas Herbarthad investigated both metaphysics and science, with a clearemphasis on the former.--Riemann was therefore able to integrate the materialistcriterion of truth, corresponding to the tradition of science,much more clearly into his essay. According to Riemann know-ledge was true if the connection of the concepts correspondedto the connection of things, which again was to be decipheredfrom the connection of the percieved phenomena [Riemann 189213,5231. --Finally, Herbart had considered the historical change ofconcepts (mainly in ontology) to have been a chain of errors[1825, 198 ff., partially cited in note [8]]. Riemann, lookingat the development of scientific knowledge, expressed a dif-ferent point of view. He declared that the relationship of newknowledge about a certain sector of reality to older knowledgeof the same sector was not necessarily that of correction anderror. He stated that modification of conceptual structure wasalso possible, which only refined the conceptual elements ofthe old system without falsification of part or al l of them [9].

    Thus, when Riemann called himself a "Herbartian in episte-mology , he meant that he had not only adopted certain centralfeatures of Herbart's epistemology, but that he had also assim-ilated the latter into the tradition of science, which enrichedit by a scientific materialist component.

    SPATIAL CONCEPTSKant's conception of fixed and a priori deducible categoriesof space and time were criticized by later philosophers of Germanidealism (Fichte, Schelling, Hegel, Schleiermacher). In thisrespect Herbart was no different. "Historical change of concepts"was used by Herbart as a strong argument against the opinionthat the fundamental concepts of reality were inborn conceptsor categories, an argument obviously directed against Kantianphilosophy. To the contrary he stressed that concepts were

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    changing products of thought [1825, 198*], including those thatserved as "forms of experience" and as such had the very functionof Kant's categories [Herbart 1829, 211.Riemann agreed with Herbart's refutation of Kant's episte-mology [1892b, 5221, although we find litt le evidence in hisexcerpts of a specific interest in Herbart's explicit discus-sions of Kant. But in one of them (R.72) we find a short polem-ical argument in that direction. In it the idea of space andtime as "empty vessels . . . in which the senses ought to pourtheir perceptions" was criticized as "a completely shallow,meaningless, and inappropriate [vbll ig gehaltlose, nichtssagende,unpassende] hypothesis" [Herbart 1824, 428*]. It seems thatRiemann thought the question was settled on this point, and wasnot very much interested in the detailed philosophical argumentthat dealt with Kant.Herbart claimed that, like al l concepts which served as"forms of experience,n spatial concepts had their origin inexperience, but then had to be shaped and developed by philo-sophical and scientific thinking. Spatial presentations areformed on this level by the perceptions of things due to the"mobility of man in his neighbourhood" [1824, 425*]. so 'I.. .the series of presentations [Vorstellungsreihen], which eventu-ally form, order, and connect themselves and in which the orderof perceptions is contained [aufbewahrt]," come into being [lo].Space and time were Herbart's starting point for generatingmore general "continuous serial forms" [continuierliche Reihen-formen] of concepts. The explanation of the latter was a verycomplicated procedure, part of the discipline of metaphysicsthat he called synechology [Herbart 1829, 110-158; Weiss 1928,50-571. Riemann's excerpts suggest that he did not botherabout specific procedures to generate "serial forms," althoughhe was interested in how all of this related to Herbart's geo-metrical thinking, because the very general idea made it possi-ble to transfer spatial concepts into a nongeometric context.Vaguely speaking, a continuous "serial form" is producedwhen a specified class of presentations undergoes a "gradedfusion" [abgestufte Verschmelzung] through which the correspond-ing presentations are ordered, so that one cannot but unitethem in a spatial mode [Herbart 1825, 192*] [il l.Consequently, Space did not exist for Herbart; insteadthere was a collection of spaces for which the modes of exis-tence were completely different [1812, 2071. His two mainexamples were the "line of sound" [Tonlinie] and the color tri-angle with blue, red, and yellow at the corners and the mixingcolors in the two-dimensional continuum in between [1825, 193*][121. Similarly he considered any thing as a "bundle [Complexionof properties," each property of which ought to be thought ofas lying in a different "qualitative continuum" [1825, 193*].

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    In this respect Herbart showed a strong aff inity to thescholastic philosophers, in particular to Nicole Oresme[Clagett 19681. Moreover, the idea of geometrizing any "thing"in that way put Herbart (like Oresme) on the very fringe ofhigher-dimensional thinking, because there was no intrinsicreason to geometrize only those "things" which could be analyzedby a collection of no more than three properties. Herbart stop-ped exactly at this barrier and claimed a limitation of spatialthinking to three dimensions 11812, 2081. In this respect hedid not go beyond scholastic philosophy.Because such general aspects of Herbart's ideas about serialforms are reflected in Riemann's excerpts (R.59, 73), it is im-portant to determine how important Herbart's transfer of space-like ideas into nongeometric concepts was for Riemann's incl in-ation to make a similar transfer inside mathematics.To start with, it is worth noting a difference of opinionbetween Riemann and Herbart. When Herbart analyzed things asbundles of properties, he emphasized that any such propertycould be understood as a "qualitative continuum" [Herbart 1825,193*] (cited in R.59). The background for the transfer of geo-metric thinking to other concepts was a very broad, nearly uni-

    versal one for him [13]. Riemann stated exactly the contraryin his inaugural lecture, namely, that "in general life" conceptsof continuous magnitudes (which would correspond to Herbart'scontinuous serial froms) were encountered only very rarely, butthat in mathematics they arose often [Riemann 1892a, 2741.The latter remark refers to a tendency in mathematics ofthe first part of the 19th century to transfer geometrical lan-guage to algebraic or analytical systems of several variables,a tendency which was at least partially known to Riemann viaGauss [Scholz 1980, 15 ff., 53 ff.] and which was independentof Herbartian philosophy. From Riemann's very clear remarkabout the dominance of mathematics in the total range of exten-sion for the concept of continuous magnitude (manifold), we maydraw the conclusion that again it was this mathematical tendencyto transfer geometric thinking to nongeometric fields which wasthe dominant background of his concept formation.This conclusion is corroborated by another difference betweenHerbart and Riemann with respect to the connotation (the "in-tension" according to Wussing [1967]) of their spatial concepts.Herbart adhered strictly to the traditional dimensional barrier(three), whereas for Riemann's concept of manifold, multidimen-sionality was one of the most essential innovations. Thisfeature of Riemann's thought was again closely linked to themathematical tendency to geometrize nongeometric algebraic andanalytic systems [Scholz 1980, 15 ff., 88 ff.], and stands instrict opposition to Herbart's opinions.Thus Herbart's direct influence on Riemann's formation ofthe concept of manifold does not appear to have been very strong.

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    However, Riemann must have found Herbart's philosophical re-flections about serial forms of presentations stimulating. Hemay have received very general hints of how to proceed in theformation of the concept, fostered by a structural similaritybetween Herbart's geometric arguments and the mathematical ten-dency "to geometrize." Moreover, he took up Herbart's idea ofan objective foundation for the formulation of spatial conceptsand sharpened it from the point of view of the scientist. Hedid so in proposing to formulate the structure of spatial con-cepts as closely as possible according to experimental physicalevidence. But the essential points of Riemann's concept ofmanifold (multidimensionality, opposition between locally simpleand globally complex behavior, separation of qualitative aspectsof extended magnitides from quantitative ones, and the separa-tion and interdependence of structures on it [Scholz 1980, 30ff., 88 ff.]) had no connection with Herbart's geometric thoughts,and Riemann had to elaborate these from the mathematical material.

    PHILOSOPHICAL STUDIES OF SCIENCEThe preceding section leads to the conclusion that Herbart'sdirect influence on Riemann's formulation of the concept ofmanifold was of rather limited importance. We must not conclude,however, that his overall influence on Riemann was similarlyrestricted. We have already found a much closer connection be-tween the two on the level of epistemology (see above), whichplayed a role in Riemann's mathematics as much as it dealt withthe question of how to proceed in research. It might be worth-while to ask whether Herbart had some influence on Riemann'sperception of the goal of mathematics or the task of mathema-

    tical research.Some answer may be formulated from the notes Riemann made(R.177) in a short evaluation of Herbart's article about philo-sophical studies [Herbart 18071. The philosophical studiesHerbart had in mind involved the elaboration of unity in diver-sity; for this he proposed working out a central concept [Haupt-begriff] in which this unity was to be reflected for each fieldof study 11807, 231, 2383.Whereas the sciences developed only central concepts linkedto their specific subject matter, it seemed necessary for Herbartto form unifying concepts transcending a specific context, andthis led from philosophical studies of the sciences to philos-ophy proper [Philosophie als eigene Wissenschaft]. The taskof professional philosophy was not just to group the most gen-eral unifying concepts worked out by the sciences neatly to-gether, but to analyze them and to resolve their intrinsicdifficulties [Herbart 1807, 2351. To do this he thought it

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    important that philosophy avoid the mistakes of both empiricism(as in the philosophies of Hume and Locke) and rationalism(Plato, Descartes, Kant, Fichte). He also rejected the concep-tion of philosophy as a dominating power "from afar," and hada much more dialectical relationship between science and phil-osophy in mind: "[philosophy] does not lie outside of otherknowledge, but constitutes itself with and in the same" [14].Consequently, philosophy and the sciences had to work handin hand. Their mutual aim was to generate many-sidedness ofeducation [Bildung] in the "higher leading class." Philosophyalways depended for its educational influence on "other scienceswhich stand nearer to the professions" [15].The generation of philosophy proper was to be achieved byspeculation [Herbart 1807, 2371, which Herbart understood in awider sense as meaning "any endeavour to make the way for tran-sitions" between concepts [1807, 2751. In a more restrictedsense speculation ought to show only the necessary connectionsbetween concepts. In any case, Herbart saw the representationof "the real" as the final goal of speculation. So philosophicalactivity (speculation) had a characteristic feature; it hadconcepts as its objects.

    This, so it seemed to Herbart, was the essential differencebetween philosophy and science which dealt with "the given."Such distinctions even held for most of mathematics up to histime: "Philosophically treated, it [mathematics] becomes partof philosophy which had to create a science of quantity [GriSBcn-lehre] for its own necessities, if one did not already exist,"[1807, 2751. For Herbart mathematics stood in even closer re-lationship to philosophy than to the rest of the sciences [16].When Riemann made his notes on this treatise [Herbart 18071,he summed up the main ideas about "speculation" with a few catch-words:. . . Speculation = endeavour [Streben] toward resolution of prob-lems. . . Demonstration of a necessary connection between conceptsA problem of further specul[-ation]..- Philosophy as a scienceGeneral character that it is generated by speculationThat it takes concepts as its object .._ (R-177) [17].

    It is likely that Riemann wrote these notes during the prep-aration of his inaugural lecture [18], which must have been avery important period for his interpretation of the methods andgoals of mathematics and mathematical sciences. It would there-fore be interesting to know why Riemann extracted just thesefeatures from Herbart's article, especially if the reason isrelated to some structural relationship between Herbart's char-acterization of philosophy and Riemann's own thoughts.

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    The similarity, in fact, of the note and Riemann's percep-tion of science and mathematics is striking. Riemann character-ized science as "the attempt to perceive nature through accurateconcepts" [1892a, 5211, which proceeded gradually through theresolution of problems resulting from contradictions in conceptsor between concepts and experience. In his opinion mathematicshad an important share in the formation, development, and ex-tension of scientific concepts, resembling very much the taskHerbart gave philosophy in relation to science. Riemann openedhis inaugural lecture with questions concerning the "possibility"and "necessity" of connections between the basic concepts ofgeometry [1892a, 2721, and he hoped that his new concept ofmanifold would be particularly helpful.Moreover the idea of clarifying conceptual structures stoodat the center of Riemann's investigations throughout his mathe-matical work, whether it was complex function theory, geometry,or integration. This was true to such an extent that one mightbe tempted to read Herbart's note as providing a characteriza-tion of mathematics as Riemann himself would have given: Ascience dealing with concepts generated to solve problems aris-ing in the attempt to gain knowledge and to clarify connectionsbetween already established knowledge ("speculation" in Herbart'slanguage).Riemann's understanding of the task of mathematics shows astriking similarity to the aims of philosophy as understood byHerbart, and the latter's characterization of the relationshipbetween science and philosophy resembles very much the relation-ship between natural science and mathematics as seen by Riemann[Scholtz 1980, 96 ff.]. In fact, Riemann's interest in Herbart'sarticle [1807] seems to have been the result of his desire toclarify his own perception of mathematics in the mirror of phil-osophy.

    This is all the more likely since a very close relation be-tween mathematics and philosophy was suggested by Herbart inthe very article summarized by Riemann. According to Herbart,mathematics became a part of philosophy if dealt with philo-sophically [1807, 2751, and at the beginning of his article, heproposed something very much along this line: The mathematicianfeels the call [Beruf] to unveil the spirit of his ingenious[geistvoll] formulas [19].It would be difficult to imagine a better characterizationof Riemann's way of doing mathematics [20].Riemann's views on mathematics seem to have been deepenedand clarified by his extensive studies of Herbart's philosophy.Moreover, without this orientation, Riemann might never haveformulated his profound and innovative concept of manifold.This represents an indirect but nevertheless effective influ-ence of Herbart on Riemann's mathematical and (in particular)his geometrical thinking.

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    CONCLUSIONSThe foregoing evaluation makes it possible to characterizemore precisely the influence of Herbart's philosophy on Riemann.The assumption that Herbart's philosophy of space had an impor-tant impact on Riemann's formulation of the manifold conceptcannot be confirmed. On the other hand Herbart's epistemologyand his ideas on the relationship between philosophy and sciencedo seem to have influenced Riemann and thus Riemann's perceptionof the task of mathematics.In the center of Herbart's philosophizing stood the practicalproblem of the education [Bildung] of man. In this respect hewas one of the German "Bildungsphilosophen" of the early 19thcentury, most of whom tended toward an idealist and dialecticalphilosophy. Starting from similar origins Herbart shared anumber of beliefs and goals with the philosophers of Germanidealism (Fichte, Schelling, Humboldt, Schleiermacher, Hegel):

    --the prominent role given to education,--the serious attempt to overcome the division between empiricismand traditional rationalism,--the central parts contradiction and change played as motiva-tion for progress in philosophy.Nevertheless, Herbart's philosophy had its own particularfeatures which distinguished it from that of mainstream Germanidealism in a number of general ways:

    --He assigned an auxil iary role to philosophy with respect tothe sciences and valued highly the empirical and the mathe-matical sciences.--His philosophy was of a moderately dialectical realism andtherefore contrasted sharply with the strictly dialecticalidealism common in Germany of the time.

    The realism appeared in Herbart's epistemology insofar asknowledge was to proceed from experience ("the given") viaconceptual clarification of the underlying reality to the ex-planation of the phenomena. Contradictions played an essentialrole in the progress of thought, but Herbart's goal was a con-ceptual system without contradiction [21]. In this respect hisphilosophy was "moderately dialect ical," in contrast to stricterdialectics which would admit contradiction, not only as adriving force for progress, but as a structural element in thedeveloped conceptual system as well.These three distinguishing features, the auxiliary role ofphilosophy, realism, and moderate dialectics, may have givenHerbart's thought a greater affinity to science and mathematics

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    than was the case for German idealist philosophy proper. Hisproposal to do mathematics "philosophically" as a science deal ingwith concepts agreed with tendencies in contemporary mathematics,and could therefore serve to stimulate Riemann in clarifyinghis ideas about mathematics. Riemann's views about the processof gaining knowledge were elaborated to a high degree as a resultof his studies of Herbart, and were influenced by the dialecti-cal elements in Herbart's epistemology. Moreover, they had aspecific influence on his mathematics. A perception of mathe-matics as a science which has to penetrate its object by a clearconceptual approach runs throughout Riemann's works like a leit-motif. Herbart's proposal to work out a central concept [Haupt-begriffl for any field of studies (see above) was realized byRiemann most clearly in complex function theory and in geometry.He started to reorganize the latter field around the concept ofmanifold, which he also thought to be appropriate for a newfoundation of physical geometry.If it could be shown that Riemann had adapted intensional(connotational) features of the concept of manifold from Herbart'sphilosophy of geometry, this would offer direct evidence of an"influx" of concrete conceptual elements into mathematics fromoutside, and the influence of philosophy on mathematics wouldhave been, in this instance, an "external factor" of conceptformation. But the situation is different. Herbart's philo-sophical conceptions of science and mathematics were assimilatedby Riemann on a very general level, influencing his ideas as tohow and in which direction mathematical research ought to pro-ceed. In this way he transformed certain features of Herbart'sphilosophy into guiding principles of mathematics which thenoperated in his own work. This was exactly what Herbart hadexpected philosophy could do in relation to science. He re-jected philosophy as "light coming from afar" and wanted todevelop it not outside but in "a thoroughly immanent relation-ship to" other knowledge [1807, 2301.It might be worthwhile to investigate this type of influenceof philosophy on mathematical and scientific research in abroader scope. In fact, Albert Lewis has already given a de-tailed study of a similar relation between a philosopher and amathematician, one very close to Herbart/Riemann in time andspace, namely, Schleiermacher's influence on H. Grassmann[Lewis 19771. Of course, the situations are different. Insofaras Grassmann was an outsider, he was hardly recognized by hiscontemporaries and remained unappreciated for most of the 19thcentury, whereas Riemann was very much in the center of mathema-tical activity. But the similarities are too striking to beoverlooked, and it would be a mistake to regard the two casesas separate and exceptional examples of the influence of phil-osophy on the orientation of mathematical research 1221. Per-haps there is a more general relationship underlying the influ-ence of Herbart on Riemann or of Schleiermacher on Grassmann.

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    In an investigation of such a relationship we must keep inmind that German philosophy of the time was not only a questionof pure thought. In an attempt to give an intellectual answerto questions arising from the confrontation of the results ofthe French revolution with the social and cultural reality ofGermany at the beginning of the 19th century [23], the problemof education [Bildunq] had been posed by German intellectuals,and it stood at the very center of contemporary philosophy,along with the idealist facing of the development of the self,the idea, or the spirit. And this had an influence on the con-tents and the structure of their philosophy.Herbart's philosophy of education, for example, was ofcrucial importance for the particular relationship philosophyshould have to the sciences and mathematics (see above). Hisdevelopmental dialectics in epistemology was part of the samecontext. And there seems to have been a common root of Herbart'sphilosophizing and that of the idealists, underlying al l theirspecific differences, in the commitment to the problem of edu-cation [Bildunq].As this commitment referred directly to the research andteaching activities of the professors at the reformed Germanuniversities, it is better to pose the question of a possiblerelationship between philosophy and mathematics during the firstpart of the 19th century (at least in Prussia and the otherGerman states) in an extended and perhaps more fundamental form:Could it be that philosophical conceptions, worked out to someextent as an attempt to master intellectually the social chal-lenges of the early 19th century, have been incorporated intomathematics (and science more broadly) as guiding principles ofresearch, thus constituting a specific link in the transforma-tion of social change into intrinsic developmental factors ofthe scientific process?Were this the case, then the relationship between Herbartand Riemann would have to be considered as a primary example ofsignificant social and philosophical influence on the mathema-tical development of the 19th century.

    APPENDIX 1: JOHANN FRIEDRICH HERBART (1776-1841)(a) Biographical Note

    Herbart studied law, philosophy, literature, and mathematicsat Jena from 1794 to 1796. At the time, intellectual circlesin Germany were heavily influenced by the experience of theFrench revolution. Attempts were made to draw consequencesadapted to the backward economic and social conditions, as wellas the philosophical and cultural context in Germany [Lassahn1978, 31 ff.]. Herbart was deeply impressed by Fichte's phil -

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    osophy and became for a while one of his main disciples [Asmus1968, 98-1071. But by the end of his studies he had developedimportant differences from the position of his teacher.After his studies Herbart worked as a private tutor for anaristocratic family at Bern (1797-1800). There he came intocontact with Pestalozzi whose pedagogical teachings he laterhelped to propagate [Weiss 1928, 1611. Upon his return toGermany, Herbart spent some time at Bremen, completed his doc-torate and his Habilitation at Gbttingen, and started to teachpedagogy and philosophy. He soon laid down the foundations forhis metaphysics and his practical philosophy, and as a result,he was offered the chair of philosophy formerly held by Kant atthe University of Kdnigsberg (in 1809).Despite his new position, Herbart did not confine himselfto philosophical teaching. He also founded a pedagogical sem-inar which he combined with an experimental school. Between1813 and 1815 Herbart was a member of the scientific deputation[wissenschaftliche Deputation] at KGnigsberg, one of three re-gional scientific advisory boards for educational reform inPrussia (the other two boards were at Breslau and Berlin;Schleiermacher was a prominent member of the latter). In thisway Herbart was active in the theory and practice of educationalreform [24].He was convinced that improvement of scientific, philosoph-ical, and cultural education was to play an important role forbroader political and scoial reforms, because he thought theor-etical education of the higher state officials to be a precondi-tion for any appropriate reform activity of the state. Moreover,education of the citizens was necessary, because only thus wouldthey be able to understand fully the meaning of state reformand the new opportunities opened by it.Herbart was a conservative reformer, as were many otherGerman intellectuals of the time, including Goethe. They ex-pected a self-reform of the monarchy's state power, which couldonly be prompted by cultural and educational activities. Forthe most part, however, they rejected the idea of social pres-sure for reform (while Herbart was in Switzerland, e.g., hecriticized the democratic radical movement harshly). InHerbart's opinion the role of the citizen was to support thereform measures applied by the state in agreement with theirown values prepared by Bildung, and thus in a spirit of "innerfreedom" [innere Freiheit].

    In 1834 Herbart took over a chair at Gattingen Universitywhere he taught philosophy and pedagogy until his death in1841. As early as the 182Os, however, the reform in Prussiaand the other German states had been confronted with the forcesof restoration and had gradually come to an end. In the 1830sand 184Os, after a short interlude of the aborted 1830 demo-

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    cratic revolutionary movement, the states of the German Confed-eration adopted a number of laws and measures to repress liberal,constitutional, or democratic aspirations. The last remnantsof liberty of the press were abolished, the secret police con-trolled public life, and even the universities were subjectedto rigorous poli tical control. In this situation reform inten-tions could no longer be articulated in public, but were forcedto take on the form of activities in conspiratorial circles orgroups which in fact were organized in several universities inGermany. Of course Herbart, following his convictions, did nottake part in these activities. The time of his involvement inreform activities had ended with the phase of state reform fromabove.The situation was aggravated in 1837 when Ernst August,King of Hannover, to which Gijttingen belonged, autocraticallysuspended constitutional rights which had been promised in adeclaration of 1833. Seven professors (including Gauss' co-worker Wilhem Weber) criticized the measure in an open letterand were suspended from their positions. This resulted inbroadly articulated student protests and lively faculty discus-sions. Herbart, however, declined to take a public positionon the question, and preferred to keep the university out ofpolit ica l struggle. This was characteristic of Herbart's in-creasing alienation from his students in the last part of hislife, resulting from his continuing rejection of reform activ-ities from below, as well as the fact that his hopes for reformsfrom above had reached a dead end.After Herbart's death (1841) his pedagogical views werediffused widely in Germany and abroad [Lassahn 19781. In dis-cussions of pedagogy he is still mentioned, but he is relativelyneglected today in philosophy. A historical reason for thisnight lie in the fact that Herbart's philosophy led him to thefringe of mainstream German idealism which shaped the philosoph-ical thinking of his time. This does not, however, explain hisrelative neglect in philosophy today.(b) Herbart's Method of Relations [Methode der Beziehungen]

    Herbart considered contradictions in experience and inalready constituted concepts as the motive force behind theircreation and modification 11829, 23, ff.]. Contradictions hadto be resolved by intellectual work resulting in the advance-ment of knowledge. Any progress was a transition from a reason(e.g., the discovery of a contradiction) to a conclusion [Grundund Folge]. The relationship between reason and conclusion,however, itself contained a contradiction. The conclusion wasto be a consequence of the reason and therefore to this extentwas contained in it. On the other hand it was to be somethingnew; otherwise it would only be part of the reason and wouldfail to give any new insight.

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    Herbart described his solution to this problem as the"method of relations" [Methode der Beziehunyen] [Herbart 1829,24 ff .; Weiss 1928, 37 ff .] . In short, he believed that thereason in the wider sense was no simple idea. The reason inthe narrower sense (designated y) had to be linked to elementsof knowledge not explicitly considered as part of it, therebygiving a system of knowledge (denoted G). The conclusion (de-noted f) then followed from G. The conclusion f formed a partof G from wh!.ch it had to be separated. This having been done,f contained something new with respect to y.

    This line of thought he held particularly useful for theresolution of contradictions in the conceptual system or be-tween concepts and experience. In this situaiton the reason'was the contradiction and the conclusion was to be the resultof an appropriate change of the concepts. Now he consideredthe dual aspect of the old concepts (9) and the more involvedcontextual situation which gave rise to the open contradiction.According to this general idea Herbart proposed a methodo-logical procedure to resolve the contradiction in which the oldconceptual system was enlarged ("blown up"), thus taking intoaccount the more involved contextual situation. From this ex-tended structure (G) the essential parts (f) had to isolatedfor a conceptual representation of the distinction which gaverise to the contradiction. Thus the contradiction was resolvedand it led to a changed conceptual system (f) as conclusion.The contradiction in the relation between reason and conclusion,Herbart argued, was thus a result of the contradictory idea thatthe reason itself is both g and G: "Therefore we say: thereason is a contradiction. To blunt the sharpness of this prop-osition means to remove all the force from the reason" [251-

    APPENDIX 2: DOCUMENTATION OF FOLIOS 140r AND 141r OFFOLDER 18 FROM THE RIEMANN NACHLASS

    (a) Folio 140rHerbart ist ausgegangen von der Untersuchung iiber dasPh(ilosophieren). Hieraus ergaben sich ihm folgende psycholo-

    gische Postulate, wozu die Erklarungen gesucht werden muBtenin der nothwendig vorauszusetzenden--und eben dadurch zu erken-nenden--Beschaffenheit und Folge der Vorstellungen:

    1. Gegensatz und Ausschliefiungskraft der Vorstellungenuntereinander. --Dieser Begriff der Vorstellungen selbst alsKr;if te (statt aller vermeintlichen Gemiithskrgfte, welche nichtsanderes sind als allgemeine Namen fiir Gruppen ;ihnlicher Phsnomene)mu@ als die Grundlage der gesammtem Psychologie angesehenwerden.-- Es gehijrt dazu das Auftreten der Zeitfolge der

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    Vorstellungen [also such . . . (unreadable, E. S.)] als Bedingungder Ichheit; weil sonst nur ein stetiges Gleichgewicht allerunter einander statt haben kijnnte.2. Anheftung des Begriffs der Negation an diejenigenVorstellungen, welche als Bilder gesetzt werden sollen. Aberder Begriff der Negation ist, so wenig wie irgend ein andererBegriff, urspriinglich in Bereitschaft: Er mu@ erst erzeugtwerden. (Das allgemeine Negieren mu6 entstehen aus denmancherlei Aufhebungen der Vorstellungen unter einander.)3. Anheftung neuer Position, oder des Seins, an die Bilderals Bilder (als des inneren Prinzips ihrer Regsamkeit).4. Auffindung dieses Seins der Bilder in der Reihe descbrigen, das da sei, und abgebildet wurde; zum Behuf der Sub-sumtion.(b) Folio 141r

    1. Bildung von Begriffen aus den Wahrnehmungen. (Abstrac-tion u. Induction. Synthesis a posteriori.)2. Erzeugung der Wahrnehmungen/des Wahrgenommenen aus denBegriffen. (Synthesis a priori.)3. Maglichst geringe VerXnderung (oder Erg;inzung) derBegriffe, wo das Wahrgenommene nach den Begriffen unmijglichoder unwahrscheinlich ist.

    In Bezug auf die Synthesis a priori zu scheiden: Woher derAntrieb? Woher die Wahrscheinlichkeit? (Biirgschaft)Der Antrieb, nach einer ErkXrung zu suchen, liegt in denvorgefundenen Begriffen.Die Wahrscheinlichkeit beruht auf der Beststigung durch dieErfahrung.H.W.Bd.4, 594. Die Methode der Beziehungen ist die Methodeder kleinsten VerZnderungen . . . . "Wenn die Erfahrung sichselbst Liigen straft, so miissen wir sie ganz verlassen, nichtaber uns riihmen, ihr so nahe als mijg lich zu bleiben" 1261.Wir brauchen uns gar nicht ernstlich mit der Erfahrung zuentzweien, wenn es sich zeigen l;isst, dass die j enige VerZinderungder Erfahrungsformen, worauf die Methode der Beziehungen unshinweist, iiberall nicht iiber die Grenzen eines solchen Fehlersder Auffassung hinwegfiihrt, den wir der Erfahrung nach gemeinerpsychologischer Beobachtung fiiglich zutrauen kiSnnen [27].In Bezug hierauf Ps. Bd.2 5139-145 "Von unserer Auffassungder Welt und den damit verbundenen Txuschungen" auszuziehen [281und die kritischen Einflechtungen auszuschildern.

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    ACKNOWLEDGMENTI am grateful for the aid I received from the Handschriften-abteilung der Giittinger UniversitXtsbibliothek, and for itskind permission to reproduce the documents of Appendix 2.I also want to thank J. J. Gray, B. Schminnes, and theeditors of Historia Mathematics for offering valuable commentson the draft of this article, thereby helping to make the finalversion more readable. Moreover, I owe special thanks toJ. J. Gray for his assistance with English style and grammar.

    NOTES1. Examples of this tendency can be found in the investi-gations of Wussing [1969] for the concept of group and Mehrtens[1979] for the case of Dedekind.2. There is other matherial that would fit the classifica-tion less well (e.g., notes to number theory on folios 182-186;summation of inf inite series, perhaps in preparation for the

    inaugural thesis, folio 208).3. "R.n" refers to Riemann Nachlass, Gijt tingen Universitatsbibliothek, folder 18, folio n.4. The * in references to Herbart indicates that the cor-responding section was studied by Riemann (the appropriate folioin question can be found in Table 1).5. "The real" [das Reale] in Herbart's usage is determinedby the fundamental categories "being, quality, inherence, andchange" [Sein, Qual it;it, InhXrenz, Ver;inderung], as developedin his ontology. It is not identical with common-sense reality.6. Auf den gegenseitigen Hemmungen der Vorstellungen untereinander beruhen die Negationen, und die Zweifel, ob such dasWahrgenommene sey oder nicht sey; endlich die Unterscheidungender Eigenschaften, denen nur ein inh;irentes Seyn, und ebendarum kein wahres Seyn zugeschrieben wird, von den Sachen, inwelche die RealitZt der Eigenschaften (des ersten Positiven)zuriick verlegt wird.Die Wanderung der Realit;it aus den Eigenschaften in dieSachen ist nur der erste Schritt zu einer weiteren Reise. Aufhaheren Bildungsstufen entsteht die Frage nach der Einfachheitder Stoffe. Wie vorhin den Eigenschaften die Sachen, so werden

    jetzt den Sachen die Elemente entgegengesetzt; diese sind nundas wahre Reale; von ihnen haben die Sache eine gelieheneRealitxt, nicht anders als vorhin d ie Eigenschaften von denSachen.Die Elemente, Feuer, Wasser, Luft, Erde--mtissen sichweiterhin die Versuche der Chemiker gefallen lassen. Nun werdenSauerstoff, Wasserstoff, Stickstoff, das Reale; hingegen Wasser

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    und Luft, vorhin Elemente, haben nur noch eine geliehene, dasheiBt, keine wahre Realitat. Jedoch such hiebey bleibt esnicht, sondern:Der Idealist findet, da6, wie die Eigenschaften, so dieSachen, die Elemente, die Grundstoffe des Chemikers, nurAnschauungen und Gedanken sind. Dahinter das Ich, welches demNicht-Ich Realit?it verleiht [Herbart 1825, 199-200; R.5gv,italics in the original].7. This is corroborated by the contents of another passageof folio 66. Here Herbart criticizes Leibniz' explanations ofchange by movements of monads, giving the counterargument thatchanges in chemistry go much deeper than would be possiblethrough changes of mere spatial relations (of monads) [Herbart1828, 130*].8. Denn die Mannigfaltigkeit der Irrthiimer iiber Substanzenund Krsfte beweis't faktisch, da.8 die Begriffe hiervon im mensch-lichen Geiste nicht vest stehn, da$ sie keinesweges Kategorienoder angeborene Begriffe sind, sondern wandelbare Erzeugnisseeines durch die Erfahrung aufgeregten, durch allerley Meinungenumhergeworfenen, Nachdenkens, welches nur dann erst in einesichere und bleibende gberzeugung iibergehn wird, wenn dieWissenschaft, Metaphysik genannt, zur Reife gelangt. Wie dieastronomische Betrachtung, die in die Weiten des Weltbaueshinausgeht, so muB such die metaphysische Forschung, welche indie Tiefen der Natur hineindringt, mancherley Revolutionendurchlaufen, ehe sie so gliicklich ist, solche Begriffe zuerzeugen, welche der Erscheinung genugthun, und mit sich selbstzusammenstimmen....1st meine Theorie [of substance--of ontology] unrichtig:so bestgtigt sie meine jetzige Behauptung, da8 die Begriffe einnoch unvollendetes Werk sind; an welchem der menschliche Geistfortdauernd arbeitet; sie best;itigt meinen Satz: da8 diemenschliche Auffassung der Welt im Werden begriffen ist.Daraus folgt dann sogleich, da@ such die T&schungen, diein diesem Werden nach einander entstehen, sehr mannigfaltig ;da@ sie den verschiedenen Bildungsstufen angemessen sind, welchesuccessiv erreicht werden; da@ sie also in kein Register, etwavon Antinomien der reinen Vernunft, sich einschlieBen lassen[Herbart 1825, 198-199; R.5qV, italics in the original].9. Die Wahrheit des Bildes ist unabhsngig von dem Gradeder Feinheit des Bildes; sie hgngt nicht davon ab, ob dieElemente des Bildes gr8Bere oder kleinere Mengen des Realenreprgsentieren. Aber die Verbindungen miissen einander ent-sprechen; es darf nicht im Bilde eine unmittelbare Wirkungzweier Elemente aufeinander angenommen werden, wo in Wirklichkeitnur eine mittelbare stattfindet. In diesem Fall wiirde das Bildfalsch sein und der Berichtigung bediirfen; wird dagegen einElement des Bildes durch eine Gruppe von feineren Elementenersetzt, so dass seine Eigenschaften theils aus einfacheren

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    Eigenschaften der feineren Elemente, theils aber aus ihrerVerbindung sich ergeben und also zum Theil begreiflich werden,SO w;ichst daraus zwar unsere Einsicht in den Zusammenhang derDinge, aber ohne dass die friihere Auffassung fi.ir falsch erkl'drtwerden miisste [Riemann 1892b, 5231.10. Ein zweyter Umstand, den wir aus unserem Verhgltnissezur AuBenwelt hervorheben miissen, ist die Beweqlichkeit desMenschen in seiner Umgebunq. Ohne diese wiirden die Anschauungender Dinge stets fiir die Dinge selbst gehalten werden; dadurchaber, da@ der Mensch einen Unterschied des Abwesenden und desGegenwXrtigen faBt, lernt er, da8 den GegenstZnden ihrErscheinen oder Nicht-Erscheinen zuf?il lig ist. Die GegenstBndebekommen, so fern sie vest stehen, such veste PlBtze in seinensich allmshlig bildenden, ordnenden, und verkniipfenden Vorstel-lungsreihen, worin die Reihenfolge der Anschauungen aufbewahrtwird [Herbart 1824, 425; R.70V, italics in the original].11. Wie der Raum auf abgestuften Verschmelzungen beruht:( ,.- ) so erzeugen sich die Vorstellungen von Zhnlichen Con-tinuen allemal unter Xhnlichen UmstZnden. Es sey demnach einegewisse Klasse von Vorstellungen so beschaffen, da@, wenn vie lederselben zugleich im Bewuptsein sind, alsdann aus ihrer Quali-t%t bestimmte Abstufungen ihres Verschmelzens erfolgen miissen:so ordnen sich unfehlbar diese Vorstellungen dergestalt nebenund zwischen einander, da@ man sie nicht anders als aufrrFiumlicheWeise zusammenfassen . . . kann [Herbart 1825, 192; R.141 , italicsin the original].12. In [1811, 1021 Herbart discussed the one-dimensionalcharacter of the line of sound by stressing the fact that in it"only a single transition" between two states [nur ein einzigerUeberqanq durch die s;-inuntlichen dazwischenlieqenden] is possible,an explanation which greatly resembles Riemann's own approachto the idea of a one-dimensional manifold [Nachlass, folder 16,folios 13-141 (compare [Scholz 19831).13. Russell's statement that Herbart had a preference forthe discrete over the continuous, and thereby had an influenceon Riemann, is plain ly misleading. See [Russell 1956, 621 andTorretti's criticism of this point [Torretti 1978, 1071.14. Diejenige Philosophie, um die es uns zu thun ist,liegt gar nicht auBer dem iibr igen Wissen, sondern sie erzeugtsich mit und in demselben, als dessen unabtrennlicher Bestand-teil ; sie had zu demselben ein ganz und gar immanentes Ver-hFiltnis [Herbart 1807, 230, italics in the original].

    15. 1st in der Gesellschaft die vielseitige Kulturzerstreut, so kann die richtige Zusammenwirkung der verschiedenenGebildeten nur dadurch gesichert werden, wenn in der hijhernleitenden Klasse vie1 Einzelne sind, deren jeder diese Mannig-faltigkeit und Bildung in sich selbst besitzt, iiberschaut,beherrscht und in der Gesellschaft zu beherrschen wei8. Aberdas innere Beherrschen der eigenen Vielseitigkeit, die letzteBestimmung und Temperatur kann nur durch die Philosophie bewirktwerden....

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    Die rechte Wirkung der Philosophie auf die Gesellschaft istdiejenige, welche durch andere Wissenschaften, die den Berufsge-schgften nXher stehen, hindurchgeht. Die Philosophie, der imGrunde kein Stoff eigenttilich zugehcrt, hat eben darum eineArt von wissenschaftlicher Allgegenwart [Herbart 1807, 300-301,italics in the original].16. Es ist namlich die Eigenthiimlichkeit dieser unsererWissenschaft: da8 sie Begriffe zu ihren Gegenstande macht.Dagegen sind die iibrigen Disciplinen vertieft im Auffassendessen, was entweder gegeben ist, oder doch gegeben werdenkbnnte. Selbst die Mathematik (denn von dem historischen Wissenist nicht nathig zu reden), so wie sie pflegt behandelt zuwerden, denkt sich ihre abstracten Formeln inuner als Formelnfiir mijgliche FBlle, und symbolisiert sehr gern ihre Functionendurch die Gestalt von Curven, wie sie iiberhaupt den Raum nurverl;iBt, urn reicher an Mitteln zur Herrschaft in ihn zuriickzu-kehren. Auch kann sie nur in dieser formellen Hinsicht von derPhilosophie geschieden werden. Philosophisch behandelt, wirdsie selbst ein Theil der Philosophie, die sich fiir ihr eigenesBediirfni8 eine GraBenlehre wiirde schaffen miissen, wenn nochkeine vorhanden ware [Herbart 1807, 275, italics in the original]17. The first part of folio 177v (folder 18 of Riemann'sNachlass) reads as follows:

    Philosophie = Untersuchung der BegriffePhil. Studien; Bd.1 pag. 556 (...)(This is the same as [Herbart 18071, E. S.)I. Phil(osophische) AnsichtenII. Speculation = Streben zur Aufldsung der Probleme (...)Nachweisung eines nothwendigen Zusammenhangs unter Be-griffenProblem weiterer Specul(-ation) als Bemiihung zwischenden Begriffen die gehErigen fibergxnge zu bahnen.III. Philosophie als WissenschaftAllgemeiner Charakter, da6 sie durch Speculation zuStande kommt; dass sie Begriffe zu ihrem Gegenstandemacht.

    The following part of the folio contains catchwords from theindex of [Herbart 18071.18. On the same page we find a preliminary draft of aletter to his father, obviously written shortly after his in-augural colloquium. II... Mein Colloquium habe ich am SonnabendVormittag gliicklich abgemacht...." It is probable that thisrefers to Riemann's inaugural lecture, since (1) it took placeon Saturday morning [Riemann 1892a, 5481, and (2) another re-mark on the same page refers to Leiden bottles, with whichRiemann worked a short while before and after that lecture.

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    19. Der Mathematiker fi ihlt den Beruf, uns den Geist seinergeistreichen Formeln zu enthiillen [Herbart 1807, 2311.20. Of course Riemann is not the only mathematician forwhom this characterization would hold (e.g., for Dedekind[Mehrtens 19791). Riemann, however, is a particularly outstand-ing example. His influence, perhaps even to a certain degreeon Dedekind, was often substantial.21. Herbart's ontology postulated a concept of "the real"free from contradictions and change.22. Thomas Kuhn's discussion of philosophical influenceson the formation of the concept of energy [1977] might indicatethat in the first half of the 19th century there was possiblyan even wider range of influence of philosophical conceptionson science.23. For the importance of the French revolution for thediscussions of the Jena circle to which the young Herbartbelonged, see [Asmus 1968, 68 ff.]; also see Appendix 1.24. As Schubrinq [1981, 191 explains, Herbart concentratedin pedagogy on individual education (as was important for pri-vate tutors) rather then on group education (as was importantfor the public school system). So his pedagogical ideas containa tension between a sense of reform and conservatism.25. Offenbar fordern wir von dem Grunde, da$, indem er dieFolge erzeugt, er selbst sich Sndert. Seine Materie sol1 sichverwandeln in die neue Materie der Folge. Hier kann nichtWahrheit an Wahrheit gekniipft werden, sondern, damit die FolgeWahrheit enthalte, muB der Grund das Geqentheil davon sein.Seine Verwandlunq dari nicht ein Verlust an Wahrheit seyn; nurein Irrthum,' der sich in nothwendiger Besserunq befindet, kannhier den Grund abgeben....Darum sagen wir: der Grund ist ein Widerspruch. DieSchZirfe dieser Behauptung abstumpfen, heipt, dem Grunde seineKraft benehmen. Denn die vollkommene Nothwendigkeit, im Denkenvorwarts zu qehen, findet sich nur da, wo das, was man schondenkt, sich selbst aufhebt [Herbart 1829, 36, italics in theoriginal].26. This is part of a fictitious citation which Herbartused to characterize possible criticism of his method of rela-tions that he anticipated from Platonists or Eleatics, etc.27. This passage is excerpted from [Herbart 1851, 594-5961.28. This refers to [Herbart 1825, 191 ff.], excerpted byRiemann on folio 59.

    REFERENCESASmUS, W. 1968. Johann Friedrich Herbart, eine p2dagogischeBiographie, Band 1, Der Denker 1776-1809. Heidelberg:Quelle & Meyer.

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    - 1970, Band 2, Der Lehrer 1809-1841, Heidelberg: Quelle &Meyer.Clagett, M. 1968. Nicole Oresme and the medieval geometry ofqualities and motions; a treatise on the uniformity anddifformity of intensities know as: Tractatus de configura-tionibus qualitatum et motuum. Madison: Univ. of WisconsinPress.Dieudonne, J. (ed.).1978. Abreg& d'histoire des mathgmatiques1700-l 900. 2 vols. Paris: Hennann.Herbart, J. F. 1796. Gber Schellings Schrift: Vom Ich oderdem Unbedingten im menschlichen Wissen. In [1964, 1, 17-331.- 1799. Erster problematischer Entwurf der Wissenslehre.In [1964, 1, 96-1101.- 1807. tiber philosophisches Studium. In [1964, 2, 227-2961.___ 1811. Psychologische Bemerkungen zur Tonlehre. In[1964, 3, 96-1181.___ 1812. Philosophische Aphorismen. In [1964, 3, 201-2141.- 1822. ijber die MUglichkeit und Nothwendigkeit, Mathe-matik auf Psychologie anzuwenden. In [1964, 5, 91-1221.- 1824. Psychologie als Wissenschaft neu gegriindet aufErfahrung Metaphysik und Mathematik, Erster synthetischerTheil. In [1964, 5, 177-4021.- 1825. Psychologie als Wissenschaft, Zweiter analytischerTheil. In [1964, 6, l-3391.~ 1828. Allgemeine Metaphysik nebst den Anfzngen derphilosophischen Naturlehre, Erster historisch kritischerTheil. In 11964, 7, l-3461.~ 1829. Allgemeine Metaphysik nebst den Anfdngen derphilosophischen Naturlehre, Zweiter systematischer Theil.In 11964, 8, l-3881.

    ~ 1851. Schriften zur Metaphysik, Zweiter Theil, JohannFriedrich Herbarts s2mrntliche Werke, G. Hartenstein, ed.,Vol. 4. Leipzig: Verlag Leopold Voss.~ 1964. Sh'mtliche Werke in chronoloqischer Reihenfolqeherausqeqeben von Karl Kehrbach und Otto Fli iqel (ErstdruckLangensalza 1899-1912). 19 vols., reprint, Aalen: ScientiaVerlag.Klein, F. 1926. Vorlesunqen iiber die Entwicklunq der Mathe-matik im 19. Jahrhundert, Vol. 1. Berlin: Springer.Kline, M. 1972. Mathematical thought from ancient to moderntimes. New York, Oxford Univ. Press.Kuhn, T. 1977. Energy conservation as, an example of simul-taneous discovery. In The essential tension, pp- 66-104.Chicago: Univ. of Chicago Press.Lassahn, R. (ed.) 1978. Tendenzen internationaler Herbart-Rezeption. Kastellaun: A. Henn-Verlag.Lewis, A. 1977. H. Grassmann's 1844 Ausdehnungslehre andSchleiennacher's Dialektik. Annals of Science 34, 103-162.

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    Mehrtens, H. 1979. Das Skelett der modernen Algebra, Zur Bil-dung mathematischer Begriffe bei Richard Dedekind. InDisciplinae Novae, Zur Entstehung neuer Denk- und Arbeits-richtungen in der Naturwissenschaft, Festschrift zum 90.Geburtstag von Hans Schimank, J. Scriba, ed., pp. 25-43.Gdttingen: Vandenhoek & Ruprecht.Riemann, B. 1892a. Gesammelte mathematische Werke und wissen-schaftlicher Nachlass, H. Weber & R. Dedekind, eds.Leipzig: Teubner.____ 189233. Erkenntnistheoretisches Fragment. In [1892a,521-5231.Russell, B. 1956. An essay on the foundations of geometry(first published, Cambridge Univ. Press, 1897). Reprint,New York: Dover.Scholz, E. 1980. Geschichte des Mannigfaltigkeitsbegriffs vonRiemann bis Poincar&. Basel/Boston: Birkhguser.

    ~ 1983. Riemanns friihe Notizen zum Mannigfaltigkeitsbe-griff und zu den Grundlagen der Geometrie. Archive forHistory of Exact Sciences (to appear).Schubring, G. 1981. Gegenstxndliche und soziale Momente desWissens als Kategorien fiir Untersuchungen der Mathematik-Didaktik. Journal fiir Mathematik-Didaktik 1, 3-36.Torretti, R. 1978. Philosophy of geometry from Riemann toPoincarg. Dordrecht: Reidel.Weiss, G. 1928. Herbart und seine Schule. Miichen: Verlag E.Reinhardt.Wussing, H. 1967. Zum historischen VerhXtnis von Intensionund Extension des Begriffs Gruppe im Herausbildungsprozessdes abstrakten Gruppenbegriffs. Zeitschrift fiir Geschichteder Naturwissenschaften, der Technik und Medizin (NTM) 4,10, 23-34.___ 1969. Die Genesis des abstrakten Gruppenbegriffs.Berlin: VEB Deutscher Verlag der Wissenschaften.