pasch's philosophy of mathematics

THE REVIEW OF SYMBOLIC LOGIC

Volume 3, Number 1, March 2010

PASCH’S PHILOSOPHY OF MATHEMATICS

DIRK SCHLIMM

Department of Philosophy, McGill University

Abstract. Moritz Pasch (1843–1930) gave the first rigorous axiomatization of projective geom-etry in his Vorlesungen uber neuere Geometrie (1882), in which he also clearly formulated the viewthat deductions must be independent from the meanings of the nonlogical terms involved. Pasch alsopresented in these lectures the main tenets of his philosophy of mathematics, which he continued toelaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivistmethodology with a radically empiricist epistemology for mathematics. By taking into considerationpublications from the entire span of Pasch’s career, the latter decades of which he devoted primarilyto careful reflections on the nature of mathematics and of mathematical knowledge, Pasch’s highlyoriginal, but virtually unknown, philosophy of mathematics is presented.

§1. Introduction. Moritz Pasch’s influence on the development of modern mathe-matics cannot be overestimated. In 1882 he presented in his Vorlesungen uber neuereGeometrie the first rigorous axiomatization of projective geometry, which has been calledthe ‘birthplace of modern axiomatics’ (Engel & Dehn, 1934, p. 133) and which earnedhim the honor of being referred to as the ‘father of rigor in geometry’ (Freudenthal, 1962,p. 619). Indeed, Pasch’s lectures exerted a considerable direct influence on Hilbert’s think-ing about geometry and axiomatics in general, as can be seen from the development ofHilbert’s lecture notes on geometry in the 1890s, which contain lengthy paraphrases ofPasch’s discussions, and also from remarks Hilbert made in his correspondence.1 This deepinfluence is not acknowledged properly in Hilbert’s seminal Grundlagen der Geometrie(1899), where Pasch is only credited in a footnote for the first ‘detailed investigations’ ofthe axioms of betweenness, in particular the axiom that became later known as Pasch’saxiom. Nonetheless, Hilbert’s brief published remarks have been reason enough for Paschbeing mentioned in almost every account of the history of modern geometry. In additionto Pasch’s impact on Hilbert, his book also exerted considerable influence on the workof Peano and of the Italian school of geometry,2 and it is discussed in detail in Russell’sPrinciples of Mathematics (Russell, 1903, pp. 393–403). But, Pasch did not only sparkthe development of modern geometry, he also lived long enough to witness its progressin the first three decades of the twentieth century.3 During all this time he was deeply

Received June 3, 20091 In a letter to Friedrich Engel from January 14, 1894, Hilbert writes with reference to Pasch’s

lectures: ‘I have learned non-Euclidean geometry solely from this book’ (quoted from Tamari(2007, p. 113)). For the development of Hilbert’s lectures on geometry, see Hallett & Majer(2004).

2 See Peano (1889b), Contro (1976), Gandon (2006), and Marchisotto & Smith (2007). I amgrateful to an anonymous reviewer for bringing to my attention the article by Gandon.

3 Moritz Pasch was born November 8, 1843, in Breslau, where he also studied mathematics withSchroter; he wrote his dissertation in 1865 in Breslau, then spent two semesters in Berlin with

c© Association for Symbolic Logic, 2010

93 doi:10.1017/S1755020309990311

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concerned with the foundations of geometry as well as those of analysis and arithmetic,and he developed a distinctive and original philosophy of mathematics.

The thesis that underlies the present paper is that Pasch’s reflections on the natureof mathematics, which he presented throughout his life, in particular in his later moresystematic accounts, are elaborations and refinements of a philosophical position that heput forward already in his famous lectures of 1882. One clear indication of this are thelater editions of these lectures in 1912 and 1926, where he had the opportunity to modifyor retract his earlier claims, but in which he only elaborated and added minor details. Thus,what we can find in Pasch is a very thoughtful and consistent approach to the foundationsof mathematics.

The two main aspects of Pasch’s philosophy are a formal stance with regard to the valid-ity of mathematical deductions and a strong commitment to an empiricist understanding ofthe basic concepts of mathematics. On the face of it, these views might appear incompatiblein various ways. Firstly, that the meanings of the mathematical terms are given empiricallymight not square with Pasch’s particular conception of deductivism, according to whichdeductions must be independent of the meanings of the terms. Secondly, the introduction ofideal elements in mathematics might stand in conflict with an empirical stance, and thirdly,empiricism might appear to be incompatible with the common view that mathematicaldeductions provide certain and necessary knowledge. (These aspects of incompatibilitywill be addressed below.) Pasch’s deductivism and empiricism are mentioned in ErnestNagel’s informative paper on the development of geometry and logic (Nagel, 1939), with-out, however, containing an account of Pasch’s attempt to reconcile them. Such an accountis also missing from Walter Contro’s detailed analysis of the axioms presented in Pasch’s1882 lectures on projective geometry (Contro, 1976), and from the recent, in parts highlyspeculative discussion in Tamari (2007).4 By drawing on Pasch’s lectures on geometry,as well as his publications on the foundations of analysis and arithmetic, and his latermore philosophical works, this paper presents Pasch’s quite unique philosophy of mathe-matics as a coherent system.5 The historical context and Pasch’s views of the relationshipbetween mathematical and philosophical investigations, which form the framework forPasch’s work, are presented in the next section. The tension between his radical empiri-cism, aimed as providing an epistemological basis for mathematics, and his goal to capturethe essence of mathematical reasoning deserves particular attention. While Pasch maintainsthat empiricism provides the best philosophical foundations for mathematics, he alsoadvances a very modern deductivist methodology for purely mathematical investigations.These views are discussed in Sections 3 and 4, respectively. Finally, Pasch’s efforts tomerge these considerations into a unified whole, which I shall refer to as Pasch’s pro-gramme, are presented in Section 5.

In presenting the reflections on mathematics and mathematical practice of a deep andclear thinker such as Moritz Pasch, who stood with one foot firm in the empiricist traditionof the nineteenth century, while vigorously striding with his other foot into the modern

Kronecker and Weierstrass, and submitted his Habilitation in 1870 in Giessen; after having beenPrivatdozent at the University of Giessen he became extraordinary professor in 1873, and wasfull professor from 1875 until 1910; he died September 20, 1930, at the age of 86.

4 For a coherent interpretation of Pasch’s mathematical work, see Gandon (2005). (Footnote addedNovember 2009).

5 Not all aspects of Pasch’s views can be dealt with in a satisfactory manner in the present paper.Some of these are mentioned in the Concluding Remarks below, and are intended to be coveredin future work.

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mathematics of the twentieth century, this paper is also intended as a contribution towarda better understanding of the radical transition mathematics underwent at the turn of thetwentieth century.

§2. Pasch’s view of mathematics. The received account of the nature of mathematicsin the first half of the nineteenth century was that given by Kant, who considered thetheorems of arithmetic and of Euclidean geometry as synthetic a priori. However, interest inthis topic was revived after non-Euclidean geometries became to be regarded as acceptableconsistent theories through the work of Riemann, Beltrami, and Klein. The presence of vi-able alternatives to Euclidean geometry cast doubt on Kant’s transcendental reasoning, andHermann von Helmholtz famously argued in the late 1860s and early 1870s that the ques-tion as to which theory of geometry describes best the space we live in should be answeredempirically.6 Around the same time reliance on mathematical intuition was also severelycalled into question by another development in mathematics, namely in function theory.In the 1860s Weierstrass lectured about the possibility of continuous functions that arenowhere differentiable and soon thereafter many other such ‘monster’ functions, which de-fied visualization and which proved commonly accepted intuitions wrong, where studied.It is against this background that Pasch formed his views on the nature of mathematics.7

One of the earliest insights into Pasch’s own development is offered in letters writtenin 1882 to Felix Klein.8 Herein Pasch mentions as influences to his views the lectures ofKronecker and Weierstrass that he attended in Berlin in 1865–1866,9 and also discussionsin the 1860s with his friend and colleague Jakob Rosanes.10 In these letters Pasch alsoexpresses his disappointment regarding the views of the few philosophers that he has read(without mentioning any names, however). Nevertheless, Pasch thought his views to be socommonsensical that he assumed them to be generally shared and he was surprised to hearof Klein’s experiences of the contrary. In print, Pasch readily points out that his views are‘by no means new’ (Pasch, 1887a, p. 129), but again without mentioning any predecessors.We find only a brief reference in Pasch (1882a, p. 17) to von Helmholtz (1876), but whetherPasch was in fact influenced by von Helmholtz, or whether he just quoted the famousscientists in support of a view that he arrived at independently or influenced by otherauthors remains an open question.

Thus, it seems that the seed to Pasch’s views on mathematics, which underlie his axioma-tization of geometry as well as his other foundational and philosophical investigations, wasplanted early in his career. As he admits without hesitation (and as will be discussed below),particular aspects of his philosophical outlook evolved over time, but on the whole Paschremained committed throughout his life to the two pillars of his philosophy: deductivism

6 von Helmholtz (1866, 1876); for a discussion, see DiSalle (1993).7 For historical overviews of the developments just sketched, see Volkert (1986) and Gray (2007).8 Letters from June 16 and 22, 1882, held at the Staats- und Universitatsbibliothek Gottingen,

Sig. Klein 11, 176, and 177. Unfortunately, not much material from the time period before 1882has been preserved in Pasch’s Nachlass at the University of Giessen.

9 This claim is repeated in Pasch’s short autobiography (Pasch, 1930b, p. 7), where he praisesboth Kronecker and Weierstrass for having taught him the necessary tools for his foundationalinvestigations.

10 In his Habilitation lecture of 1870 Rosanes states that ‘in more recent times, one haspredominantly switched over to von Helmholtz’s so-called empiricist theory, according to whichspace is nothing more than a concept that has been abstracted from experience’ and also mentionsLocke as an proponent of an empiricist theory of space (Rosanes, 1871, p. 8; emphasis in original).

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Table 1. Layers of mathematical and philosophical investigations

Investigations Goals

Research (‘rough’) New mathematical resultsMathematical

Foundational (‘delicate’) Stem concepts and propositions

Foundational Core concepts and propositionsPhilosophical

Pre-scientific Necessary conditions, skills, etc.

and empiricism. Pasch published a systematic account of his views, which I turn to next,only in the last two decades of his career.

In order to accommodate his deductivism and empiricism into a coherent picture, Paschdistinguishes between different layers of mathematical and philosophical investigations,which are characterized by different aims and methodologies (see Table 1). According tothis picture, mathematical investigations take place at two distinct layers. The first one,which Pasch describes as ‘rough’ (‘derb’) mathematics, comprises the usual work that isdone by mathematicians in order to obtain new results (Pasch, 1918a, p. 230).11 The bulkof mathematical research falls into this category, and it is worth mentioning already at thispoint that as a practicing mathematician Pasch was well aware of the distinction betweenhow mathematics is presented and how it is actually done (more on this later).

The second layer of mathematical work is foundational in character and it involves care-fully working out the fundamental concepts and propositions of a discipline and showinghow the entire discipline can be built up from them. Pasch refers to this part of mathematicsas ‘heikel’ (Pasch, 1918a, p. 230), which is translated here as ‘delicate’, but could also mean‘finicky’ and ‘touchy’. His own axiomatization of projective geometry (Pasch, 1882a) andhis introduction to analysis (Pasch, 1882b) are examples of such investigations. Delicatemathematics is guided by the difficult demand for a ‘scrupulous completeness of the trainsof thought’ (‘unbedingte Vollstandigkeit der Gedankengange’) and is motivated by an ‘urgefor pure knowledge’ (‘entspringt [ . . . ] dem Drange nach Erkenntnis uberhaupt’) (Pasch,1924a, p. 36). Such investigations aim at an axiomatic presentation of a mathematicaldiscipline, which Pasch calls a stem (‘Stamm’), consisting of stem concepts and propo-sitions (Pasch, 1882a, pp. 74, 98).12 On their basis a mathematical theory can be built updeductively, and as long as they are not given any philosophical grounding, Pasch alsorefers to them as ‘hypothetical’ (Pasch, 1917, p. 185) or ‘mathematical’ (Pasch, 1924a,p. 43).

Once a mathematical foundation of a discipline has been given, the philosophical taskarises of determining the meanings of the mathematical terms and of giving an accountof their applicability to the world. In other words, a ‘substructure’ (‘Unterbau’) has tobe provided that supports and grounds the mathematical theory (Pasch, 1917, p. 185).13

For these philosophical foundations different approaches are possible, and Pasch mentions

11 See also Pasch (1924a, p. 35).12 In Pasch (1924a, p. 16), Pasch refers to stem propositions also as ‘basic propositions, axioms,

postulates’. For more on Pasch’s choice of terminology, see below.13 See also Pasch (1924a, p. 42).


rationalist, a priori, and empiricist accounts as alternatives (Pasch, 1926c, p. 138). Forreasons that will be discussed later (Section 3), Pasch himself decided to pursue a radicalempiricist approach. The details of Pasch’s efforts to connect the philosophical substruc-ture to the purely mathematical foundations are discussed under the heading of ‘Pasch’sprogramme’ below (Section 5).

Finally, a second layer of philosophical investigations is concerned with uncovering theconditions, skills, and so forth, that are necessary for employing the basic concepts and car-rying out the investigations at the higher levels. Pasch refers to this layer as investigationsregarding the ‘prescientific origins’ or simply the ‘origin’ (‘Ursprung’) of mathematicsand thinking in general (Pasch, 1924a, p. 40), and he identifies as its fundamental conceptsthose of ‘thing’, ‘proper name’, ‘event’ (in particular that of ‘naming a thing’), ‘collectivename’, ‘earlier’ and ‘later’ events, ‘immediate following’, and ‘chain’ of events (Pasch,1924b, p. 234).14 Pasch’s investigations at this layer might be characterized, borrowing anexpression of Hilbert, as a deepening of the foundations of human knowledge.15

The investigations at each of the top three layers can be pursued independently of theconsiderations pertaining to a lower layer, which allows for the division of ordinary andfoundational research as well as the division of mathematical and philosophical labor. Asa consequence, mathematicians can ignore the questions regarding the origins and theapplicability of mathematics altogether, and most often they do.16 For Pasch, however,a complete picture of mathematics requires an account of each of these four layers and oftheir interconnections. To emphasize and illustrate this organic, hierarchical structure Paschemploys terminology that evokes the picture of a tree of mathematics: On the one hand,he refers to the philosophical foundations as a ‘Kern’, which is rendered here as core, butcould also be translated as ‘pip’ or ‘kernel’, that consists of core concepts and propositions(‘Kernbegriffe’ and ‘Kernsatze’) (Pasch, 1916).17 On the other hand, the mathematicalfoundations of a discipline are called a ‘Stamm’, translated here as stem, which but couldalso be rendered as ‘stalk’ or ‘trunk’, that consists of stem concepts and propositions(‘Stammbegriffe’ and ‘Stammsatze’); in accordance with this botanical metaphor, the do-main of philosophical inquiry that is common to all sciences is referred to as an area ofroots (‘Wurzelgebiet’) in Pasch (1924a, p. 34).

Failure to notice Pasch’s distinction between a (mathematical) stem and a (philosophical)core, and indiscriminate reference to both stem propositions and core propositions as‘axioms’ has led to misinterpretations and disputes in the literature. For example, Kline(1972, p. 1008) mentions that some of Pasch’s axioms have empirical origins, while Torretti(1978, p. 211, and footnote 49) explicitly disagrees with this assessment and claims thatPasch considers all axioms to be empirically grounded.18 Since also Nagel does not addressPasch’s crucial distinction between a core and a stem (Nagel, 1939, pp. 193–199), it hasalso been missed by many later commentators who relied heavily on Nagel’s account.

14 See also Pasch (1927), Pasch (1930a), and (Pasch, 1980, p. 16). The search for such origins cancertainly be traced back to Kant, but also some of Pasch’s colleagues addressed such questions,for example, Dedekind (1888, p. 336), and Veronese (1894, pp. 1–2).

15 See Pasch’s discussion of Hilbert (1922) and Hilbert (1923) in Pasch (1924b, pp. 236–240).16 See Pasch (1912, p. 204), Pasch (1924a, p. 43), and Pasch (1927, p. 123).17 Core propositions are also referred to as ‘primitive stem propositions’ in Pasch (1924a, p. 16

[1915]), and in Pasch (1924b, p. 232) the core is referred to as a ‘ “natural” stem’.18 A similar claim is made in Boniface (2004, p. 133).

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In the course of his career, which spanned over 60 years, Pasch devoted his attentionincreasingly to the deeper layers of mathematical and philosophical investigations, de-scribing his aim as ‘getting as far as possible to the beginnings’ (Pasch, 1926b, p. 166).Pasch’s earliest publications were a few short research articles, after which he broughtout two books in 1882, both of which are concerned with foundational mathematicalwork and are interspersed with philosophical reflections. Soon after Pasch’s lectures ongeometry, his Einleitung in die Differential- und Integralrechnung (Pasch, 1882b) ap-peared.19 But, Pasch was not able to develop the foundations of analysis as deeply ashe had intended, and he tried to remedy this in Grundlagen der Analysis (Pasch, 1909)and Veranderliche und Funktion (Pasch, 1914). Only after his retirement from teaching foralmost four decades at the University of Giessen,20 Pasch found the time to write on morephilosophical topics. This led to numerous articles and the collections Mathematik undLogik (Pasch, 1919, 1924a), Mathematik am Ursprung (Pasch, 1927), and Der Ursprungdes Zahlbegriffs (Pasch, 1930a).21

According to his strong conviction of the existence of a tight connection between ‘cor-rectness of linguistic expression and correctness of thinking’ (‘Sprachrichtigkeit und Denk-richtigkeit’) (Pasch, 1930b, p. 6),22 Pasch always struggled to find the most appropriateterminology for expressing his ideas. For example, while he distinguished between ‘basic’and ‘stem’ concepts and propositions in Pasch (1882a, pp. 74 and 98), he began referring tothe former as ‘core’ in Pasch (1916, p. 276), remarking that his original terms ‘Grundsatze’and ‘Grundbegriffe’ were often understood in a different sense than he intended. Changesin terminology also reflect changes in Pasch’s way of thinking. For example, the distinctionbetween ‘rough’ and ‘delicate’ mathematics (Pasch, 1918a, p. 230) was first introduced asone between ‘consistent’ (‘konsistent’) and ‘disputable’ (‘strittig’) mathematics 2 yearsearlier (Pasch, 1916, p. 275). Similarly, the distinction between ‘proper’ and ‘improper’mathematics that Pasch introduces in Pasch (1914, pp. 153–157) was later reformulated asone between ‘perfect’ and ‘imperfect’ mathematics (Pasch, 1918a, p. 230).23 In later yearsPasch also urged to employ different names for mathematical notions and their empiricalcorrelates, on the grounds that their conceptual differences can be easily overlooked ifthey are both referred to by similar names, and he suggested the terms ‘location’, ‘path’,‘segment’, ‘bowl’, and ‘plate’ (‘Stelle’, ‘Weg’, ‘Strecke’, ‘Schale’, ‘Platte’) as names forthe empirical conceptions of point, line, straight line, surface, and flat surface (Pasch, 1917,p. 187).24

Pasch was very well aware of the tentative character of axiomatic presentations and hecontinuously tried to improve on his previous work by publishing lists of corrections to

19 The preface of Vorlesungen (Pasch, 1882a) is dated ‘March 1882’, while that of Einleitung (Pasch,1882b) is dated ‘May 1882’.

20 See Pickert (1980, pp. 49–57) for a list of the courses taught by Pasch in Giessen.21 Interestingly, more publications by Pasch appeared in the two decades after his retirement than

before.22 Pasch elsewhere describes the aim of mathematics as ‘the most complete clarity of thought and

of their linguistic expressions’ (Pasch, 1924a, pp. 39–40); for a practical example, see Pasch(1887b, p. 132), where he introduces new terminology that allows for ‘more precise and shorterformulations’.

23 The distinction between perfect and imperfect mathematics will be discussed in connection withthe notion of decidability in Section 4, below.

24 See also Pasch (1930a, p. 19) for Pasch’s use of ‘Rotte’ instead of ‘Reihe’, which he used in Pasch(1909, p. 7).


earlier publications and slightly changing formulations even in reprints.25 Pasch’s attitudetoward foundational work is expressed quite tellingly his review of a book by Dingleron the notion of logical independence in mathematics, subtitled Also an Introduction toAxiomatics (Dingler, 1915). Here Pasch criticizes the author for reprinting an obviously un-finished article without further revisions and for not seriously trying to present a completeset of core propositions (Pasch, 1916, p. 276). Pasch concludes that the book would needfurther ‘patient work’ before being able to yield concrete results from the accumulated rawmaterial. In addition, Pasch demands higher standards regarding the ‘exactness of one’sthought and expression’ and a more thorough self-criticism especially from somebodywho writes an introduction to axiomatics. There are good reasons to believe that he didhold himself responsible to such standards.26

§3. Pasch’s empiricism. Pasch’s version of empiricism, the main points of which Iwill try to outline in this section, differs in important respects from the views held by hiscontemporaries, but bears some resemblance to the views of Berkeley, Locke, and Hume.27

In contrast to the question as to which geometry is the ‘right’ description of space, whichwas the driving force behind von Helmholtz’s form of empiricism, Pasch’s main concernwas the nature of the fundamental concepts of mathematics. A satisfactory account of this,according to Pasch, must answer questions regarding the applicability of mathematics aswell as the epistemology and certainty of mathematical knowledge. In accord with mythesis that the main elements of Pasch’s philosophy can be found already in his lectureson projective geometry, I shall begin the discussion with Pasch’s remarks on the nature ofgeometry.

In the opening sentence of his Vorlesungen uber neuere Geometrie (Pasch, 1882a, V),Pasch laments that the empirical origins of geometry have not been consistently broughtout in the recent treatments of this discipline that tried to meet the increased standardsof rigor, and he announces that his lectures aim at carrying out such a project. Shortlyafter this pronouncement Pasch justifies his point of view by claiming that the successfulapplications of geometry in daily life and in science are based on the fact that the geometricconcepts originally conformed exactly with empirical objects, and that only later theywere ‘covered by a network of artificial concepts’ to foster the advancement of theoreticaldevelopments. By restricting himself from the start to empirical concepts only, Paschintends to retain the character of geometry as a natural science.28 A few pages later herepeats his resolve of steadfastly holding on to the empiricist standpoint, according to

25 This can be seen, for example, in the additions to the 1912 edition of his lectures on projectivegeometry and the various (seemingly overly pedantic) corrections to previous publications thathe adds in later works. Just to mention a few, Pasch (1909) contains corrections to Pasch (1882a)on pp. 117–188 and to Pasch (1882b) on p. 120; corrections to Pasch (1912) are listed in Pasch(1914, VI) together with further corrections to Pasch (1909). A number of small changes in thetext can be found in the versions of Pasch (1894) reprinted in Pasch (1909), Pasch (1919), andPasch (1924a).

26 Pasch’s publications, as well as his autobiographical reflections and the descriptions of hischaracter and work ethic by people who knew him personally confirm this; see Pasch (1930b,p. 10), Dehn (1928), Engel & Dehn (1934), and Tamari (2007).

27 See Jesseph (1993, pp. 44–87) and Pressman (1997); see also John Stuart Mill’s A System of Logic(Mill, 1851), and Harre (2003).

28 The view that geometry is a natural science is frequently echoed by Hilbert, see Hallett & Majer(2004, pp. 66, 197, 266, 504).

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which ‘geometry is seen as nothing else but a part of natural science’ (Pasch, 1882a, p. 3).Thus, Pasch presents his work from the outset as providing philosophical foundations forprojective geometry, in addition to purely mathematical ones (see Table 1, above). Thiscombination, which is rather unusual in its extent for a mathematical treatise, may havecome as a surprise and possibly also as an irritation to readers who did not share the aimsand methods of Pasch’s conception of mathematics and his philosophical project.29

Pasch’s original move, which characterizes his version of empiricism and sets it apartfrom that of his contemporaries, is to take empirical concepts as the starting point for arigorous development of mathematical theories. This involves two main steps: First, thestem concepts of a discipline have to be developed from the empirical core concepts, andsecond, the remainder of the theory has to be based on the stem concepts alone. Note,that in order to guarantee the empirical character of mathematics as a whole, its theoremsmust inherit the epistemological status of the axioms; it is at this point where Pasch’sdeductivism becomes fundamental for establishing his empiricism.

As a general and essential criterion for the choice of core concepts Pasch holds that theyshould be able to explain how the mathematical concepts originated or at least how theycould have originated (Pasch, 1917, p. 190).30 Moreover, they should be as few as possibleand express the simplest content possible (Pasch, 1894, p. 24). Pasch also insists that thebasic terms of a mathematical theory can neither be defined nor can they be reduced to otherconcepts, but that we can only understand them through reference to appropriate physicalobjects (‘den Hinweis auf geeignete Naturobjecte’) (Pasch, 1882a, p. 16). In particular, theprinciple of duality in projective geometry, that is, the fact that the basic terms in the stempropositions can be interchanged systematically while yielding again valid propositions,is taken by Pasch as evidence that these propositions cannot be regarded as definitionsof the basic concepts (Pasch, 1914, p. 143). This stands in direct contrast to the modernunderstanding of axiom systems as implicit definitions of its primitive terms.31 Thus, ingeometry Pasch introduces points as those objects that cannot be further divided withinthe limits of observation determined by the best tools that are currently available to us.He also rejects the common view that lines must be ‘ “imagined” as being infinitely ex-tended’, since such a demand does not correspond to any perceptible objects (Pasch, 1882a,p. 4); instead, Pasch takes the notion of (finite) line segments as a core concept.

In addition to the demand that the basic objects of geometry should be observable, theymust satisfy some further restrictions in order to be usable. For all practical purposes,configurations of physical geometric objects (i.e., figures or diagrams), Pasch explains,must be such that, on the one hand, the observer is relatively close to them, and on the otherhand, that their parts are sufficiently close to allow for an immediate grasp of their rela-tionships (Pasch, 1882a, pp. 18–19). As a consequence, one can have immediate evidencethat these relationships hold only within a relatively small, bounded region of space.32

29 That readers might be irritated is mentioned, for example, in Tamari (2007, pp. 77, 194–195). Twoexamples: Russell speaks of Pasch’s ‘empirical pseudo-philosophical reasons’ (Russell, 1903, p.393) and in a recent commentary Majer mentions some ‘curiosities’ that characterize Pasch’sapproach (Majer, 2004, p. 104).

30 For similar remarks, regarding the axiomatization of arithmetic, see Pasch (1924a, p. 16 [1915]).31 That Pasch did understand axioms to implicitly define the primitives is claimed in Tamari (2007,

ii, p. 6, and 96). But, compare the footnote in Pasch (1920, p. 145), in which Pasch explicitlydenies such an interpretation; see also Gabriel (1978).

32 For similar views on these fundamental assumptions Pasch refers to Riemann (1854, p. 266),Klein (1871, pp. 576 and 624), and Klein (1873b, p. 134).


In a similar vein, Pasch also notes that general terms (universals) are introduced only withreference to a finite number of particular objects (Pasch, 1914, p. 3).

For Pasch, the further development of a mathematical discipline proceeds from obser-vations to propositions. In geometry, repeated observations of concretely given figuresyield simple relations between the basic concepts, some of which are formulated as basicpropositions, from which all other propositions of geometry follow. For example, two ofPasch’s basic propositions are: ‘I. Between two points one can always draw one and onlyone segment’ and ‘VI. Given any two points A and B, it is possible to choose a point C ,such that B lies within the segment AC ’ (Pasch, 1882a, p. 5). Given the empirical referentsof the primitive terms, Pasch notes that these propositions do not hold in general, butthat they are subject to certain restrictions. In order to draw a segment between any twopoints, these points must be sufficiently apart from each other, while the points A and Bof basic proposition VI must be sufficiently close to each other to allow for the actualconstruction of the third point. These conditions are met in the usual diagrams or mentalvisualizations that accompany mathematical investigations, but they must be made explicitand kept track of in the deductive development of geometry. Theorems that depend onthe above basic propositions, are thus also subject to restrictions. For example, also theconstruction expressed in theorem 8, which states that ‘Given two points A and B on a line,it is always possible to choose a point C on that line, such that C lies between A and B’(Pasch, 1882a, p. 10), and which is proved using the above axioms, cannot be appliedindefinitely often (i.e., the ‘always’ must be taken with a grain of salt).

While Pasch does not give specific arguments for his empiricist standpoint in his earlywritings, he does provide an argument in Pasch (1914, pp. 138–139), which is based onthe applicability of mathematics. In order to apply mathematical propositions to the world,the concepts that occur in them must be related to things that occur in experience, whichis straightforward if they are understood to refer to empirical notions from the outset.If, however, mathematical concepts are not understood as referring to empirical objects,in which case Pasch calls them ‘hypothetical concepts’, their applicability rests on twosets of hypotheses: First, the axioms themselves are purely hypothetical, and second, theassociation between mathematical concepts and their empirical correlates is hypothetical,too. The position that Pasch describes here bears strong similarities to that of a hypothetico-deductive account of mathematics, which must be augmented by ‘coordinative definitions’to be applied.33 From Pasch’s empiricist standpoint, however, these two kinds of hypothe-ses present a detour that does not add any benefits, so that the empiricist approach is simplerand thus to be preferred. In Pasch (1917, pp. 185–186) Pasch remarks that ‘hypotheticalgeometry’ is completely independent from physical objects (‘Naturgegenstanden’), whichbecomes completely obvious if the terms ‘thing of the first, second, and third kind’ are usedinstead of ‘points, lines, planes’, as was suggested by Hilbert (1899). From a mathematicalpoint of view this way of proceeding is unobjectionable for Pasch, but it leaves the relationto figures and applications unexplained. More generally, he maintained that despite thefact that the problem of applicability had been widely discussed from a nonempiriciststandpoint no satisfactory solution had yet been given.

A second argument for empiricism is presented in Pasch (1922, pp. 3–4) and Pasch(1924a, p. 44). Here Pasch notes that different viewpoints regarding the nature of geom-etry, for example, that it is ‘a pure creation of human thought’, find their expression in

33 See Reichenbach (1957, p. 14) or Nagel (1961, p. 93); Hilbert’s account of the application ofmathematical theories is also similar.

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introductory textbooks. However, a closer look at these expositions also reveals that noneof them remains completely consistent in presenting geometry from a single point of view.For example, a ‘body’ may have been defined as a part of space that is delimited on allsides, but it is later said to be moved, despite the fact that a part of space is not somethingmovable (Pasch, 1924a, p. 44). Without going into further details Pasch argues that it isimpossible to purge all allusions to experience from any introduction to geometry, andthus, that a coherent presentation should treat geometry as an empirical science. The lackof a textbook that consistently pursues the empiricist standpoint is explained by the factthat much more work needs to be done to lay bare the prescientific foundations that such apresentation would require.34 Nonetheless, Pasch also recommends teaching geometry inschool by starting with empirical notions, since they are what seems to come most naturallyto beginners (Pasch, 1909, pp. 134–135).

§4. Pasch’s deductivism. In addition to exploring the philosophical foundations ofmathematical concepts, Pasch was also interested in capturing an ideal of mathematicalreasoning and in providing a general criterion for mathematical rigor. He shared this goalwith his contemporaries Frege and Hilbert. A related concern of Pasch regards the clarifica-tion of the role of intuition in mathematical reasoning and he engaged in brief discussionswith Study and Klein on this issue.35 Pasch’s later investigations led to a careful analysisof the nature of proofs and to discussions of the notions of decidability, consistency, andmathematical discovery. As in the previous section, I shall begin with Pasch’s earliestreflections on these matters.

Pasch informs us in Pasch (1918a, p. 231) that he arrived at his views on deductiononly while writing the Vorlesungen uber neuere Geometrie (1882). Just as the openingsentence of these lectures introduces the reader to the empiricist background of the book,the next sentence expresses the second cornerstone of Pasch’s philosophy of mathematics:the view of geometry as a science that obtains its results ‘by purely deductive means’(Pasch, 1882a, V). What Pasch means by this is that regardless of the content that isintended to be conveyed by the basic propositions, once these have been put forward norecourse to perceptual experience should be necessary for the further development of thetheory (Pasch, 1882a, p. 17). He expresses this very modern deductivist stance even moreexplicitly in the most often quoted passage from his lectures as follows36:

In fact, if geometry is genuinely deductive, the process of deducing mustbe in all respects independent of the sense of the geometrical concepts,just as it must be independent of figures; only the relations set out be-tween the geometrical concepts used in the propositions (respectivelydefinitions) concerned ought to be taken into account. (Pasch, 1882a,p. 98; emphasis in original)

34 Pasch mentions Thaer & Lony (1915) as a valuable attempt in this direction.35 On Eduard Study, see Hartwich (2005); I intend to discuss the interactions between Pasch and

Klein in a subsequent paper.36 Quotations of this passage can be found, for example, in Nagel (1939, p. 197), Kennedy (1972,

p. 133), Torretti (1978, p. 211), Shapiro (1997, p. 149), Boniface (2004, p. 134), Detlefsen (2005,p. 251).


This passage has been referred to as the ‘birthplace of modern axiomatics’37 and isthe basis for Tamari’s reference to Pasch as the ‘father of modern axiomatics’ (Tamari,2007, title), and Freudenthal’s remark that ‘[t]he father of rigor in geometry is Pasch’(Freudenthal, 1962, p. 619). How quickly Pasch’s conception of mathematical deductionbecame widely accepted can be gleaned from the fact that 8 years later Klein mentionsPasch as espousing an ‘almost generally held view’, according to which in geometricconsiderations one has to rely only on axioms without making any use of intuition (Klein,1890, p. 571). Pasch never grew tired of emphasizing again and again the importance ofthis understanding of deduction, which he also referred to as ‘the genuine mathematicalmethod’ (Pasch, 1918a, p. 228) and as an ‘imperative’ (‘Gebot’) for mathematical research,which is completely independent of any position regarding the philosophical foundationsof mathematics one might want to adopt (Pasch, 1917, p. 188). As he repeated over threedecades after his lectures on geometry were published, mathematical proofs must remainvalid if the basic concepts are replaced throughout ‘by any concepts or by meaninglesssigns’ (Pasch, 1914, p. 120).38 He refers here to this method as ‘a formalism, that has tobe carried downright to the extremes’ in the development of mathematics (Pasch, 1914,p. 121; emphasis in original), and concludes emphatically: ‘This formalism is the lifeblood(‘Lebensnerv’) of mathematics’ (Pasch, 1914, p. 121).39

Replacing meaningful terms by variables, for example, changing ‘There are points’ to‘There are αs’, is the key to formalization, according to Pasch. He emphasized that geomet-ric arguments must remain valid even if the geometric terms are replaced by code names(‘Decknamen’) like ‘P-thing, G-thing, and E-thing’ (Pasch, 1918a, p. 231).40 In general,for a mathematical proof to be rigorous it must rest only on propositions that allow suchsubstitutions and whose inferences remain valid under such transformations (Pasch, 1926c,p. 263). This analysis leads Pasch to distinguish between ‘material words’ (‘Stoffworter’)and ‘joins’ (‘Fugemittel’) (Pasch, 1926c, pp. 243 and 263). As Pasch explains, the formerare meaningful terms that denote concepts, which are the material (‘Stoff ’) of the propo-sition, like ‘two’, ‘points’, ‘segment’, and ‘endpoint’. The joins constitute what is neededto connect the material words in order to express relations between the denoted concepts,and they include what are now called the logical parts of expressions.41 They are calledthe ‘scaffolding’ (‘Gerust’) of a stem in Pasch (1924a, p. 11 [1915]), and in Pasch (1894,p. 21) he explains that in order to carry out deductions one only needs to understand ‘thoseparts of language that are common to all domains of thought’ (‘Denkgebieten’). This allowsPasch to reformulate his understanding of deduction as follows:

The mathematical proof has nothing to do with the meaning of the ma-terial words; it depends ultimately only on the joins and thus presents apure formalism. (Pasch, 1926c, p. 263; emphasis in original)

37 See Engel & Dehn (1934, p. 133) and Pickert (1982, p. 271).38 At this point Pasch does not distinguish terminologically between words and concepts, that is,

between linguistic entities and their meanings. He addresses this distinction, however, in Pasch(1926c). See also Pasch (1909, p. 1).

39 This remark is echoed in Pasch (1926c, p. 263).40 For similar considerations, see also Dedekind’s letter to Lipschitz, July 27, 1876 (Dedekind,

1932a, p. 479), and Hilbert’s letter to Frege, December 29, 1899 (Frege, 1980, p. 40).41 It appears that Pasch’s distinction between material words and joins is intended to distinguish

nonlogical from logical components of expressions in a natural language.

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Formalization, as the most reliable touchstone for the validity of proofs, can be dis-pensed with if one is very careful, but this is very difficult, Pasch warns, as the gap inEuclid’s first proof illustrates (Pasch, 1926c, p. 140). Nevertheless, Pasch acknowledgesthe usefulness of diagrams in mathematical practice, as will be discussed below (p. 107).Pasch (1914, pp. 121–137) discusses at some length four historical case studies of math-ematical errors made by Ampere, Cauchy, Dirichlet, and Hasse, which he traces back toa lack of rigour in the development. Such rigor can be achieved through formalization,which for Pasch is a powerful technique to ascertain the logical validity of arguments.As such, Pasch’s notion of formalization does not involve the presentation of mathematicalreasoning in a symbolic language like Peano’s or in a completely formalized languagelike Frege’s. Pasch explicitly distanced himself from these approaches and promoted for-malization only to the extent that it remained compatible with ordinary mathematicalpractice.42

Pasch realized that his understanding of deduction requires to address the question ofwhat counts as a mathematical proof, and in a letter to Frege from 1894 he expressed hissurprise of finding how rarely this topic had been seriously investigated.43 In a lecture onthe value of mathematical education delivered in the same year, Pasch pointed out thatmathematical proofs serve two main goals. Originally, they were a means for ‘discoveringnew properties of figures and numbers’, but later they were also employed for examiningthe ‘logical dependencies’ among propositions (Pasch, 1894, pp. 23–24). The second pointis vividly illustrated by Pasch’s own investigations in Pasch (1882a), in particular thediscussions of various equivalent axiomatizations in Section 1. Pasch did not consider thestudy of mathematical inferences as a subject matter of mathematical research per se, butas a matter of independent and general importance properly belonging to the domain ofphilosophy (Pasch, 1914, p. 33). As is evident from the correspondence with Frege, Paschshowed great interest in Frege’s work, but he also remarked that due to his age and theheavy demands on his time he was not in a position to familiarize himself with Frege’snotation.44 Nevertheless, Pasch undertook his own investigations of the notion of proof inorder to give an account of the necessary conditions for valid mathematical inferences thatapply to informal arguments as well as to those presented in a formal language. While hisinvestigations remained only in the fledgling stages, Pasch expressed the hope that theymight lead to a ‘renewal of logic’ and that ‘the indicated path will lead to the main featuresof a logic that does justice to the accomplishments of mathematics’ (Pasch, 1918a, p. 232).The position that Pasch arrived at is that

[i]t is part of the essence of pure deduction that every proof can be‘atomized’, i.e., resolved into steps of certain kinds, or that it consistsof a single such step. (Pasch, 1917, p. 189)

In his ‘Begriffsbildung und Beweis in der Mathematik’ (1925) Pasch illustrates and dis-cusses in great detail how the Aristotelian syllogistic form Barbara, that is, the deductionof ‘All As are Cs’ from the premises ‘All As are Bs’ and ‘All Bs are Cs’, can be atomizedinto 16 individual steps. According to Pasch’s analysis, each of these steps in the deduction

42 See Pasch’s letter to Klein, October 19, 1891; held at the Staats- und UniversitatsbibliothekGottingen, Sig. Klein 11, 184.

43 Pasch’s letter to Frege, February 11, 1894 (Frege, 1980, p. 103).44 Letter from Pasch to Frege, January 18, 1903 (Frege, 1980, p. 105). The preserved correspondence

with Frege consists of seven letters from Pasch in the period from 1894 to 1906.


is the reformulation of the same content in other words, or the dissection of the originalcontent while retaining only a part of it, or the combination of the contents of previoussteps, or a definition.45 Pasch concludes that the most basic inferential steps must be suchthat one can decide, by a general method in a finite amount of time or of steps, whetherthey are valid or not. He also discusses questions that he considers to be undecidable, suchas whether a given proof can be rendered gap free and whether a given formula is derivablefrom a set of assumptions. Pasch appeals here to a notion of decidability that he attributesto Kronecker and which he recognized only during his work on Grundlagen der Analysis(1909) as being of fundamental importance.46 Pasch (1914, pp. 153–157) distinguishesbetween ‘proper mathematics’, which makes only use of decidable notions, and ‘impropermathematics’, which does not, but which is much more common. To avoid speaking of‘improper’ mathematics, Pasch later changed the terminology to ‘perfect’ and ‘imperfect’mathematics (Pasch, 1918a, p. 230).

In his 1894 lecture mentioned above Pasch remarks that neither Euclidean nor non-Euclidean geometries contradict any facts of experience, but that nevertheless thesesystems could still be inconsistent (‘einen inneren Widerspruch enthalten’), ‘because ex-perience only refers to approximated usability, which is quite compatible with certaininconsistencies’ (Pasch, 1894, p. 31). He also maintains that explicit and complete proofsfor the consistency of both geometries are still lacking, and suggests that such proofscould be based on analytic means, which would settle the question at least for thosewho consider the consistency of analysis as necessary (Pasch, 1894, pp. 31–32). Morethan 20 years later Pasch took up the issue of consistency again in a lecture ‘Uber in-nere Folgerichtigkeit’ (1915), which was published in Pasch (1919). Here he introduces aclassification of inconsistencies, two of which are ‘internal’ to a theory, while the othertwo concern applications of theories. An inconsistency of the first level occurs withina single sentence or between two given sentences. Since they involve only a finite setof sentences, such inconsistencies are ‘decidable’ (granting the investigator a sufficientlylong life and a big enough memory). An inconsistency of the second level, however, is onebetween consequences of a given stem, which are in general infinite in number, and Paschnotes that we do not have any general process by which we could decide whether such aninconsistency obtains or not, since this would involve the determination of all consequencesof a set of axioms. As a method for establishing consistency Pasch explains how a givenset of meaningful propositions can be ‘formalized’, resulting in an ‘empty stem’, whichin turn can be ‘realized’ by replacing the meaningless symbols by meaningful concepts,yielding a ‘filled stem’ (Pasch, 1926a, p. 11 [1915]). If a realization of a formalized stem isconsistent, Pasch argues, then the original stem is also consistent. In modern terminology,Pasch here describes the notion of relative consistency proofs. To ‘arithmetize’ a stem,which often depends on a ‘felicitous idea’, amounts to showing its consistency relativeto arithmetic.47 Pasch defends the standpoint that arithmetic is indeed consistent, whilethe consistency of any other mathematical discipline must be established by a proof.48

45 See also Pasch (1924a, p. 38), where Pasch refers to his earlier discussions of proofs in Pasch(1909), Pasch (1912), and Pasch (1914).

46 See Pasch’s discussion of the notion of decidability in Pasch (1914, pp. 153–157), Pasch (1918a),and Pasch (1927, pp. 88–93).

47 Pasch presents this terminology as if it were his own. References to Weierstrass, Kronecker, andKlein are conspicuously missing; see Klein (1895) and the discussion in Boniface (2007, p. 332).

48 Kronecker’s views on the natural numbers seem to loom in the background of this discussion, butPasch never mentions them explicitly (see also Footnote 9).

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To show the consistency of arithmetic one would have to show that an axiomatization ofarithmetic itself is consistent, or, in other words, this should follow from arithmetic itself.49

Pasch argues that this is indeed the case by appealing to the intimate connection betweenarithmetic, thought, and language. The source of arithmetic, which finds expression in itscore propositions, Pasch contends, is necessary for thought in general and its constituentsare so extremely primitive that we are not consciously aware of them. We have committedourselves to their content when we made experiences and fixed them in language. It followsthat these propositions and their consequences are binding for us, which justifies, forPasch, our belief in their consistency (Pasch, 1924a, p. 17 [1915]). It is worth pointingout that Pasch does not mention any empirical considerations in his discussions of theconsistency of mathematical theories, except in the argument for grounding the consistencyof arithmetic.50

The value of the deductive method in mathematics, for Pasch, is that it excludes all arbi-trariness in proofs and thus renders them unassailable, which, together with the empiricalevidence for the core propositions, is the basis for our ascribing the ‘highest level of reli-ability’ to mathematics (Pasch, 1882a, p. 100). Notice how, as a thoroughgoing empiricist,Pasch does not speak of the necessity of mathematics, but only of its reliability.51 For allpractical purposes mathematical knowledge is as good as certain. Although strict adherenceto the deductive method in mathematics might lead to more long-winded expositions, it hastwo further advantages for mathematical practice, according to Pasch. Firstly, proofs thathave been carried out without any appeal to the meanings of the nonlogical primitivesoccurring in them are reusable, in the sense that replacing the terms in the assumptions insuch a way that they become true statements automatically also yields true statementsfor the conclusions if the terms are replaced accordingly (Pasch, 1882a, pp. 98, 100).In this way new mathematical results can be obtained ‘in a purely mechanical fashion’without having to repeat the derivations (Pasch, 1914, p. 120). Secondly, a deductivepresentation of a domain can be exploited to determine which concepts and propositionsare necessary or dispensable for the theoretical development of the discipline (Pasch,1882a, p. 100).52

As mentioned above, Pasch never suggested that ordinary mathematics should be carriedout in a formal system and he seriously doubted the feasibility of such an undertaking.Instead, formalization is only a technique, albeit a very powerful one, for ascertaining therigor of deductions. Already in the first edition of his Vorlesungen (1882) Pasch notes thatit is admissible and useful to think about the meanings of the geometric terms during adeduction, but that as soon as this becomes necessary the incompleteness of the deduction

49 Any reference to Hilbert (1900) or Hilbert (1905) are again conspicuously missing from Pasch’sdiscussion.

50 See Pasch (1894, p. 17) and Pasch (1909, p. 134), which are referred to in Pasch (1917, p. 185).It is also noteworthy that Pasch does not discuss Dedekind’s nor Peano’s axiomatizations ofarithmetic (Dedekind 1888; Peano 1889a); indeed, in Pasch (1927, p. 90) he remarks that there isno generally accepted set of core propositions for arithmetic. It is possible that he did not acceptDedekind’s notions of system and mapping as being empirically grounded, and that he objectedto Peano’s use of a purely symbolic language.

51 Pasch only rarely speaks of the truth of the core propositions, for example, he refers to them as‘basic truths’ (‘Grundwahrheiten’) in a talk to a general audience (Pasch, 1894, p. 21). I thinkPasch would agree to Einstein’s famous remark that ‘As far as the laws of mathematics refer toreality, they are not certain; and as far as they are certain, they do not refer to reality’, quoted from(Hempel, 1945, p. 17).

52 See also Pasch (1924b, p. 233).


is revealed. He mentions that it is actually a ‘common conception’ that theorems shouldfollow logically from axioms (Pasch, 1882a, p. 99), but that his remarks on mathematicalrigor are nevertheless not uncalled for, since this demand remains unfulfilled more oftenthan not, even in publications that deal explicitly with the foundations of a mathematicaldiscipline.53 As possible reasons for this discrepancy Pasch mentions the frequent use ofdiagrams or other perceived or imagined pictures that accompany a deduction. But, Paschdoes not at all reject the use of diagrams in mathematical practice. In fact, he acknowledgestheir efficiency for representing the relations that are expressed in the assumptions orconstructions of a geometric proof ‘in an intuitive way’ (‘in anschaulicher Form’) (Pasch,1882a, p. 43). On the one hand, such a representation makes it easy to survey and bringback to memory the relevant relations, and, on the other hand, it stimulates the ‘inventivetalent’ (‘Erfindungskraft’) and it is thus a fruitful tool for discovering new relations andconstructions (Pasch, 1882a, p. 43). Indeed, for Pasch the ‘creative trains of thought’ bywhich mathematicians advance to new knowledge ‘must not and cannot’ be rendered com-pletely in formal terms (Pasch, 1926c, p. 142). Diagrams are necessary for understandingthe core propositions of geometry, Pasch maintains, since the latter express observationsmade on simple figures. Moreover, every inference can be confirmed by a figure, althoughthe figures themselves do not justify the inferences. How diagrams can be misleading isillustrated by Pasch’s discussion of the first proof in Euclid’s Elements.54 Pasch brieflyconsiders the possibility of admitting inferences that are based on figures, but he quicklydismisses it with the comment that one would hardly succeed to delineate clearly whichinferences would be acceptable and which would need to be justified by recourse to earlierstatements made in the proof (Pasch, 1882a, p. 45). In a similar vein Pasch also notesthat the use of familiar terms in mathematical discourse can be misleading, since theyevoke, often unconsciously, many associations that are not logical consequences of theaxioms (Pasch, 1882a, p. 99). Again, these considerations are considered unobjectionableby Pasch when it comes to the discovery of new geometric truths, where all means canbe applied that lead to the end, but not for the verification and presentation of the results(Pasch, 1882a, p. 99).55

§5. Pasch’s programme. Given the distinction Pasch makes between the mathemati-cal and philosophical foundations of a discipline, the general problem arises of connectingthese two. In Pasch’s case, this problem is exacerbated by the fact that he opted for thephilosophical foundations to be grounded empirically, instead of, for example, in a Platonicrealm of mathematical objects. Thus, his overall framework is put under stress by thetension between his empiricism and his deductivism. Pasch’s programme, intended to easethis tension, consists in finding adequate ways of combining these two standpoints and indeveloping deductive theories from empirical cores.

Since the reader of Pasch’s Vorlesungen (1882) might easily miss the general aim andstructure of his approach, the development can appear unmotivated and needlessly

53 Pasch (1917, p. 188) mentions Du Bois-Reymond (1882) as an example.54 Pasch’s discussion seems to have contributed much to the popularization of the critique of Euclid’s

proof, so much that it even has been referred to as the first instance of such a critique, for example,in Friedman (1985, p. 462).

55 Here Pasch distinguishes clearly between what has later been called the ‘context of discovery’ andthe ‘context of justification’ (Reichenbach, 1938, pp. 6–7); see also his ‘Forschen und Darstellen’(Pasch, 1919), in particular p. 35. For a contemporary discussion of this matter, see Hersh (1991).

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cumbersome. In fact, the stem concepts and propositions of projective geometry,56 whichone would expect to find at the beginning of the book, are only introduced in section10, after almost 100 pages of long-winded deductions and definitions from the empiricalcore concepts and propositions that Pasch starts out with. A reader interested in studyingprojective geometry may well wonder what the first 100 pages are all about. It is only in‘Grundfragen der Geometrie’ (1917) that Pasch explicitly presents his overall conceptionof geometry (presented in Section 2, above) and discusses his method of ‘extending themeanings of concepts’. This is taken up again 5 years later and discussed in detail withreference to the ‘deep contrast’ (‘tiefen Gegensatz’) between ‘physical geometry’ and‘mathematical geometry’ (Pasch, 1922, pp. 362–363).

Let us take a look at how Pasch presents these matters in his 1882 lectures on geometry.Here he describes the relation between mathematical theories and their empirical founda-tions by stating that ‘[m]athematics sets up relations between the mathematical concepts,which should correspond to facts from experience’ (Pasch, 1882a, p. 17). This makes itsound as if all mathematical propositions have direct empirical correlates. However, whilethis characterization might well have been an ideal that Pasch had in mind at the time,it does not square with his own way of developing the axioms of projective geometryin his lectures. Pasch’s approach is captured more accurately in his later, more nuancedreflections, in which he only speaks of correlates that have been developed from empiricalpropositions.

Once the empiricist has completed the substructure, he can attach to itthe theory that I referred to as mathematical geometry without changingthe wording. He would then, whenever one speaks of points understandit as ‘mathematical points’, a concept which has been developed fromthe physical point in the substructure. (Pasch, 1924a, p. 43).

Thus, Pasch’s strategy for bridging the gap between empirical and mathematical con-cepts can be characterized as follows: Start with empirical core concepts and proposi-tions, and develop theorems through definitions and deductions, which can be used ascorrelates of the mathematical stem propositions of a particular discipline. This enterprisemay involve two different kinds of moves: (a) the lifting of empirical restrictions, and (b)the definition of new concepts that extend previous ones. These are illustrated in whatfollows.

Consider the statement ‘Between any two points on a line segment there is anotherpoint’. Taken as an empirical statement, it is false. Due to the limits of our perceptionand the fact that points must be extended to be observable, two points might just beso close to each other that there is not enough space to fit another point between them.Thus, if the statement is to be understood as expressing a core proposition about empiricalpoints, it must be augmented with the proviso that the points in question be sufficientlyapart from each other. As a mathematical proposition, however, the above statement canbe accepted without any further restrictions on the relative locations of the points. Thus,we can obtain mathematical statements from empirical ones by simply dropping certainadditional conditions, which is one way of connecting a mathematical theory with itsempirical substructure.

56 They are 22 in total: 8 for line segments, 4 for the plane, and 10 for congruency; the latter arecommon to Euclidean and non-Euclidean geometry.


However, not all mathematical propositions can be obtained by simply removing restric-tions that are necessary for empirical ones. If this is the case, Pasch resorts to a techniquehe refers to as ‘extending the meanings of concepts’ (‘Begriffserweiterung’) (Pasch, 1882a,p. 64) which he employs, for example, for points, lines, and planes in Pasch (1882a)and for numbers in Pasch (1882b). To motivate this technique, Pasch also mentions twoextensions of concepts from the history of mathematics: The notion of number originallymeant only positive rational numbers, but was extended at some point in history to includealso negative numbers, while the notion of power was originally used only for naturalnumbers as exponents, but was also extended to include negative and rational numbers in asimilar way (Pasch, 1882a, pp. 40–41). Analogously, Pasch introduces the concept ‘thing’for concrete objects, but extends it in Pasch (1909, p. 20) to include sequences and in Pasch(1909, p. 94) to include infinite sets.57

Since the method of extending the meanings of concepts plays a crucial role for develop-ing deductive mathematical theories from an empirical core, I will present next how Paschproceeds to extend the meanings of ‘numbers’ and then of ‘bundle of lines’ and ‘points’.Pasch extends the meaning of the concept ‘number’ in his Einleitung in die Differential-und Integralrechnung (1882) to include also irrational numbers on the basis of Dedekindcuts (Dedekind, 1872). He begins by introducing the word ‘number’ as referring onlyto positive whole numbers and their quotients, that is, to nonnegative rational numbers,and assumes that for these the relations of equality, greater, and less than, as well as theoperations of addition, subtraction, multiplication, division, and exponentiation are known.After deriving a few basic theorems from these assumptions, Pasch notices that not everynumber can be represented as the power of another number (e.g., that there is no number xin the domain, such that x3 = 25). Following Dedekind, he considers all numbers whosenth power is less than a given number a as forming a ‘group’ (‘Gruppe’), which Paschcalls ‘number segment’ or just ‘segment’. Then he notes that for some numbers a and nthere is a least number that does not belong to the corresponding number segment (e.g.,for a = 25 and n = 2 this least number is 5), but that for others there is no such leastnumber (e.g., for a = 25 and n = 3). Pasch calls those number segments with a least upperbound rational and the others irrational, and proceeds to define the relations of equality,greater, and less than, as well as the operations of addition, subtraction, multiplication,and division for number segments. On the basis of these definitions he argues that allexpressions that involve these notions for numbers also hold of number segments, bothrational and irrational. With regard to powers, Pasch shows that, unlike in the case ofnumbers, every number segment can be represented as the nth power of another segment,and he shows that the powers remain well defined and obey the familiar laws not onlyfor every rational number segment, but also for every irrational one. This allows Paschto notice that the computations with segments completely subsume the computations withnumbers, but also go beyond them, since they allow for the unrestricted application ofthe inverse operation of taking the power. Once the theory of number segments (which is‘more complete’ (Pasch, 1882b, p. 11), since subject to fewer restrictions than the theoryof numbers) is adopted, the term ‘number’ plays no particular role any more, since it canbe replaced throughout by ‘rational number segment’. This observation motivates Pasch to

57 Another example for the extension of concepts concerns the notion of limit, see Pasch (1918b).I am grateful to an anonymous reviewer for bringing this to my attention.

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dispense with the old meaning of ‘number’ and use this term for number segments instead,so that now we can speak of ‘rational’ and ‘irrational numbers’.58

In sum, Pasch’s extension of the meaning of the term ‘numbers’ proceeds in three steps:first, it is taken to refer only to nonnegative rational numbers; then, rational and irrationalnumber segments are introduced, the former of which correspond to numbers; finally, theterm ‘number’ is applied to number segments in general, which also allows to speak of‘rational’ and ‘irrational numbers’. Further extensions of the domain of numbers to includenegative numbers, zero, infinity, and imaginary numbers are also mentioned by Pasch, buthe does not present them in detail.

To show the usefulness of the newly introduced concept of number, Pasch discussesthe measurement of straight lines. Empirical measurements, he notes, can only be madeup to a certain limit of accuracy, but mathematics aims at establishing general rules thatare independent of limitations of what can be observed (‘Beobachtungsverhaltnisse’). Thiscan be achieved by admitting also irrational numbers, since then no knowledge of anyparticular threshold of accuracy is required (Pasch, 1882b, p. 13).59

It is informative to notice the striking contrast between Pasch’s and Dedekind’s presenta-tions of the introduction of irrational numbers. While Dedekind uses abstract set-theoreticterminology and considers the real numbers to correspond to points on a line, Pasch usesmore concrete terminology in his approach and takes the limitations of our empiricalinteractions with lines as the starting point, distinguishing between the calculation of thelength of a line and its measurement. Moreover, while Dedekind clearly distinguishesbetween a cut and its corresponding number, Pasch redefines the term ‘number’ to referto cuts, but, as was not uncommon at the time, he does not distinguish carefully betweenthe term ‘number’ and the concept of number. Since Pasch obviously would not want toassert that a number has infinitely many elements, he must restrict his number talk to onlycertain properties of cuts, but he completely avoids to address this issue. Pasch also doesnot comment on the problem that the uncountability of the irrational numbers might posefor his empiricist approach.

The latter difficulty points at a more general issue of Pasch’s programme, namely theexact specification of the means that he regards as admissible for the development of newconcepts from given ones. Since Pasch’s attitude is not revisionist, he must be open inprinciple to accept the results that are obtained by any method used in mathematics. Oneway of showing the compatibility of mathematical practice with his empiricist standpointis to find ways of achieving the same results by licensed methods.

In the case of projective geometry, it was accepted practice to introduce ideal pointsas ‘points at infinity’ where parallel lines meet.60 Such a definition, however, does notconform to Pasch’s empiricist standards, because infinity is not an empirical notion, and sohe set out to introduce these objects by other means. In his lectures on projective geometryPasch extends the meaning of ‘bundle of lines’ and ‘point’ (Pasch, 1882a, pp. 33–46).On the basis of the notions of points and lines, the latter of which he defined using the

58 In an unusually opinionated review for the Jahrbuch fur Fortschritte in der Mathematik Pasch’sredefinition of ‘number’ was severely criticized for being circular by Hoppe, who insists that theconcepts of number segment and irrational number should be kept apart (Hoppe, 1882).

59 Pasch always remained sceptical with regard to the applications of irrational numbers outside ofmathematics. After explaining how the square root of 2 arises from considerations regarding thediagonal of a unit square, he writes in Pasch (1909, p. 99): “It remains open as to whether everyirrational number corresponds to a problem outside of analysis.”

60 See Torretti (1978, p. 111).


core concept of line segment, Pasch initially defines a bundle of lines, as the collection oflines that meet in a common point. He then proves that for lines e, f , and g the relation‘g belongs to the bundle e f ’ can be defined using the property of being coplanar, butwithout making any reference to the point in which e and f meet. Thus, this relation canalso hold between lines e, f , and g, if e and f have no point in common, and Paschtakes it as the defining characteristic for an extended notion of bundle of lines. Also forthese bundles Pasch shows that they are determined by any two lines that belong to them.Moreover, if two lines in a bundle meet in a point P , then all lines of the bundle meetin that same point, so that some bundles can be said to correspond to a unique point,namely P , and these are the ones that were formerly referred to as bundles and are nowcalled proper bundles. Other bundles, however, may contain lines that do not intersect,such that there is no obvious relation between these bundles and particular points, andthey are called improper bundles. Finally, after having defined these notions and provedsome properties about them, Pasch extends the meaning of the term ‘point’ to refer tobundles of lines instead (just as he extended the meaning of the term ‘number’ to referto number segments). Those bundles that correspond to a point in the original sense arethen referred to as proper points, while the others are called improper points. Thus, theextended meaning of the relation of ‘line l goes through point P’ is that of ‘line l belongsto bundle of lines P’. The advantage of this change in terminology is that previous axiomsand theorems about points and lines remain valid also under the extended meaning. Forexample, ‘For any two points there is a line that that goes through both of them’ is alsovalid if ‘point’ refers to a bundle of lines. In addition, now also statements that were falseunder the original restricted understanding of points become true, if understood as referringto points in the extended sense, for example, ‘Two lines in a plane always have a pointin common’. Pasch’s improper points had previously been treated as ideal elements inprojective geometry and Torretti describes Pasch’s approach of introducing these elementsonly on the basis of ostensive concepts and empirically justifiable axioms as his ‘mostremarkable feat’ (Torretti, 1978, p. 213).

Nowadays we would describe Pasch’s method of extending the meaning of conceptsby saying that the term ‘point’ is given two different interpretations: while it originallyreferred to points, it is later taken to refer to bundles. But, Pasch does not yet possessa conceptually clear distinction between syntax and semantics, and it does not seem tocome naturally for him to speak of different interpretations of a given term, in particular,since ‘point’ is a meaningful term, unlike, say, a mathematical variable. Thus, he speaks ofsubstituting one concept for another in a proposition, changing the meanings of concepts,or replacing concepts by meaningless signs.61

In geometry, the notion of continuity also goes beyond what can be developed on anempirical basis and it indicates the conceptual gap between empirical and the mathematicalgeometry. Pasch carries through the development of projective geometry to allow for the in-troduction of coordinates through the construction of rational nets. Thus, these coordinatesremain limited to rational values. Nevertheless, he notices that ultimately only an analytictreatment of geometry in terms of real coordinates yields the customary notions of pointsand lines, which Pasch qualifies with the adjective ‘mathematical’. He briefly considersthe possibility of adding an axiom of continuity, but dismisses it on the grounds that itwould not be empirically justified and opts for a version of the Archimedean axiom instead

61 See also Pasch’s notions of ‘formalization’ and ‘realization’ of a stem, discussed in Section 4,above.

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(Pasch, 1882a, p. 126).62 However, given that all his empirically based constructions aresubject to limitations, mathematical points allow for more fine-grained distinctions thanempirical points do; in other words, every empirical point corresponds to an entire sequenceof mathematical points. This phenomenon is referred to as the ‘inexactness of geometricconcepts’ and Pasch emphasizes that ‘the transfer of a diagram into numbers and thereturn from the results of a calculation to the diagram cannot be carried out with the samedegree of exactness’ (Pasch, 1882a, p. 200).63 Nonetheless, Pasch remarks that also themathematical points (if appropriately defined) satisfy the stem propositions of projectivegeometry. Given that he considers real numbers themselves to be grounded in empiricalcore concepts, this does not seem to pose a serious problem for his philosophy, but itconfirms his assertion that geometry presupposes arithmetic (Pasch, 1922, p. 5).

Pasch commented that one of the goals he pursued in his lectures on geometry wasto show that a reduction of parts of geometry to empirical notions was possible in prin-ciple (Pasch, 1887a, p. 130),64 and it appears that many of his contemporaries acceptedthis reduction. For example, in the article on geometry by Weber and Wellstein in theEncyclopedia of Elementary Mathematics (1905), Pasch’s book is discussed in the firstchapter on the fundamental notions of geometry (Weber & Wellstein, 1905, pp. 25–27).After formulating a critique of the idealization processes that are intended to lead fromthe empirical raw material of geometry to its abstract objects, the authors ask whether itis possible to build up an intuitive geometry, which they call ‘natural geometry’, withoutrecurring to these idealizations, and they note that an affirmative answer to this question ispresented in Pasch’s lectures on projective geometry, ‘this beautiful book that everybodymust have read, who is more interested in intensive rather than extensive knowledge ofgeometry’ (Weber & Wellstein, 1905, p. 25).

§6. Concluding remarks. We have seen how Moritz Pasch formulated the corner-stones of his philosophy of mathematics in his two books of 1882 and continued to developand refine his views in numerous, more and more philosophical publications throughouthis life. Pasch’s philosophy is quite unique in combining a strong empiricism, according towhich the meanings of mathematical terms should be based on observable physical entities,with a deductivist view, according to which the validity of mathematical inferences doesnot depend on the meanings of the terms. These seemingly incompatible views are broughttogether in Pasch’s conception of different layers of philosophical and mathematical inves-tigations, and ‘Pasch’s programme’, which aims at building up correlates of mathematicalaxioms from an empirical basis. Since Pasch’s philosophical ideas originally appeared onlyas interspersed remarks in his mathematical textbooks and were elaborated in more detailonly in his later articles it has been difficult to grasp and appreciate his philosophy of math-ematics as a whole. This might be part of the reason for the general lack of awareness of hisideas, which Pasch himself noticed (Pasch, 1926b, p. 166). Another reason might be that

62 See Ehrlich (2006, p. 6) and Greenberg (1993, p. 125), who writes that ‘[t]he full significance ofArchimedes’ axiom was first grasped in the 1880s by M. Pasch and O. Stolz’.

63 Pasch reminds the reader in Pasch (1882b, pp. 13 and 39) that every number that is used in practiceor that arises from observations or measurements can only have a limited degree of exactness, andhe adds in a remark on p. 188 that the inexactness of geometric concepts had been discussed byKlein already in 1873 (Klein, 1883). This is repeated at other occasions, for example, Pasch(1887a, p. 130) and Pasch (1912, p. 203).

64 See also Pasch (1912, p. 203).


his methodological considerations regarding the nature of mathematical deduction werequickly absorbed by his contemporaries like Peano and Hilbert, whose own contributionssoon overshadowed those of Pasch.

In the present overview I was able to deal with a number of issues in Pasch’s philosophyonly cursorily, but they nevertheless deserve further and more detailed exploration. Amongthese are: the origins, principles, and limits of Pasch’s empiricism together with a com-parison with the British empiricists and Pasch’s contemporaries; Pasch’s ideas about themost basic constituents of mathematical reasoning and thinking in general; Pasch’s notionof intuition, in particular in comparison with Klein and Study; the role of definitions, inparticular of implicit definitions, in Pasch’s works; Pasch’s analysis of logical inference,in particular in comparison with Frege and Hilbert; Kronecker’s notion of ‘decidability’that became more and more important for Pasch and his distinction between ‘proper’and ‘improper’ mathematics; finally, the reception and influence of Pasch’s work in thenineteenth and twentieth centuries. I hope to have shown in the present paper that Pasch’shighly original philosophy of mathematics is worthy of further study and of becominggenerally known, both in its own right and as a part of the mathematical and philosophicalcontext in which modern mathematics emerged.

§7. Acknowledgments. I would like to thank, first and foremost, Michael Hallett forhelpful comments on previous versions of this paper. In addition, I am also grateful forremarks and comments by an anonymous reviewer of this journal, Greg Frost-Arnold,Jeremy Heis, Paul Rusnock, as well as audience members at the Annual meeting of the As-sociation of Symbolic Logic (Irvine, CA), the PhiMSAMP-3 conference ‘Is mathematicsspecial?’ (Vienna, Austria), the HOPOS meeting 2008 (Vancouver, BC), the Winter 2008meeting of the Canadian Mathematical Society (Ottawa, ON), and the Winter meeting ofthe Association of Symbolic Logic (Philadelphia, PA). Last, but not least, I would like tothank Dr. Rudolf Thaer for generously sharing his knowledge about Pasch with me. Workon this paper was funded by Social Sciences and Humanities Research Council of Canada(SSHRC). Translations are by the author, unless noted.

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