Abstract This paper offers an exposition of Husserl’s mature philosophy ofmathematics, expounded for the first time in Logische Untersuchungen andmaintained without any essential change throughout the rest of his life. It isshown that Husserl’s views on mathematics were strongly influenced byRiemann, and had clear affinities with the much later Bourbaki school.

Keywords Husserl Æ Logic Æ Mathematics Æ Manifold Æ Riemann ÆBourbaki Æ Platonism Æ Zermelo-Russell Paradox.

Introduction

In Chapter X of the first volume of Logische Untersuchungen1. Husserldiscusses briefly his affinities with and differences from the views on logic ofpast philosophers. He clearly states that Leibniz, Bolzano and Lotze, in thisorder, are those philosophers with whose views he coincides the most. Inparticular, this affinity is grounded on the common understanding that logicand mathematics are very close relatives. In this sense, Husserl’s views arealso related to those of his contemporaries Frege and Hilbert. They are allLeibnizians with regard to logic and its relation to mathematics, but theintellectual grandsons of the great Leibniz interpret differently their Leib-nizian heritage. As is well known, for Frege and his intellectual inheritors, like

G. E. Rosado Haddock (&)Department of Philosophy, University of Puerto Rico, Rio Piedras, PR 00931-1572, USAe-mail: grosado@uprrp.edu

1 Logische Untersuchungen I (from now on abbreviated as LU), 1900–1901, Husserliana XVIII,1975 (hereafter cited as Hua XVIII).

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Husserl Stud (2006) 22:193–222DOI 10.1007/s10743-006-9010-y

Husserl’s philosophy of mathematics: its originand relevance

Guillermo E. Rosado Haddock

Received: 21 March 2006 / Accepted: 21 July 2006 /Published online: 19 December 2006� Springer Science+Business Media B.V. 2006

Whitehead and Russell, mathematics (in Frege’s case non-geometricalmathematics) was reducible to logic, in the sense that all its concepts could bedefined in terms of logical concepts and all its theorems derived from logicalaxioms. They were, thus, logicists, as this term is currently understood in thefoundations of mathematics.

Husserl, like Hilbert, was no reductionist either to logic or to set theory, asclearly expressed already in 1890 in a letter to Carl Stumpf,2 and although hehad begun his investigations on the foundations of mathematics under theinfluence of Franz Brentano’s mild psychologism, as attested by his profes-sorship’s dissertation Uber den Begriff der Zahl3 and its expansion anddevelopment Philosophie der Arithmetik,4 published at the beginning of 1891,already about 1890 he began to distance himself from such early views. Such achange was the primary reason why Husserl did not publish the second vol-ume of his early work, even though that volume, in contrast to the first one,should have dealt with the logical foundations of arithmetic. The book thatwould result from such a development of his thought could not be seen as thesecond part of Philosophie der Arithmetik. That book was going to be pub-lished a decade later under the title Logische Untersuchungen. Husserl’smisgivings with respect to his earlier views received a decisive impulse fromthe reading, in the early 1890’s, precisely of the philosophers men-tioned above, namely, Leibniz, Bolzano and Lotze, as well as of—of allpeople—David Hume, as clearly attested in the Husserl Chronik5 and in hisIntroduction to the Logical Investigations.6

But the abandonment of his former psychologistic leanings was not the onlyreason for not writing a second volume of Philosophie der Arithmetik, avolume which as projected, was in any case to be concerned with non-psychologistic foundations of arithmetic. At least since 1890 Husserl hadexpanded his interests in foundational affairs to the whole of mathematics, asattested by the publication of his (sketches of) papers in Studien zur Arith-metik und Geometrie.7 The influence of his friend and colleague GeorgCantor, and of Bernhard Riemann, Felix Klein and others was already verystrong. Husserl’s mathematical studies, especially as a student and assistant ofKarl Weierstrass (although he was also a student of Leopold Kronecker), hisfriendship with Georg Cantor, also a former student of both Weierstrass andKronecker and a mathematician of overwhelming philosophical interests, andhis study of Riemann and the whole tradition based on the latter’s work wasthe other component, besides the Leibnizian, which contributed to thedevelopment of Husserl’s views on logic, mathematics and their relationship.

2 Reprinted in Studien zur Arithmetik und Geometrie (from now on abbreviated as SAG),Husserliana XXI, 1983 (hereafter cited as Hua XXI).3 Uber den Begriff der Zahl, 1887, reprinted as an appendix to Husserliana XII, 1970.4 Philosophie der Arithmetik, 1891, Husserliana XII, 1970 (Hereafter cited as Hua XII).5 Karl Schuhmann, ed., Husserl-Chronik, 1977, p. 26.6 Introduction to the Logical Investigations, 1975, pp. 35–38.7 See footnote 2 above.

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Riemann was, thus, Husserl’s other intellectual grandfather, the one he didnot have in common with his intellectual cousin and rival Gottlob Frege. Inthis paper, after expounding Husserl’s views in the extremely importantChapter XI of the first volume of Logische Untersuchungen, which we willcomplement with other Husserlian works, we will briefly comment on thisimportant source of Husserl’s philosophy of mathematics as well as on a verypopular current view of mathematics with clear affinities to Husserl’s, namely,that of the Bourbaki school.

Preliminaries

In the first five sections of Chapter XI (§§62–66) Husserl is concerned with thenature and objectivity of science. Although these sections are not part of hisphilosophy of mathematics, but just preliminaries, they contain some valuabledistinctions and insights of Husserl, which can help in making a thoroughassessment of Husserl’s importance for contemporary non-continental phi-losophy. At the beginning of §62 Husserl reminds8 us of the three senses inwhich one can speak of the unity of a science, namely, the anthropological andpsychological unity of acts of thought, which is subjective and of no interesthere, the unity of the domain of the science, which is objective and objectual (inGerman: gegenstandlich), and the unity of the truths about that domain, whichis objective but not objectual. These two objective domains, with which Husserlis here concerned are inseparable but should not be confused with one another.The connections between truths do not coincide with the connections betweenthe objectualities about which those truths are truths. Thus, the objectual unityof a science does not coincide with the unity of its truths. In an act of knowl-edge, e.g., in physics we are concerned with something not merely objective butobjectual. In some sense, says Husserl,9 truths enter the scene as an idealcorrelate of the act of knowledge. ‘‘To the connections of knowledge,’’ Husserladds,10 ‘‘there correspond ideally the connections of truths,’’ since connectionsof knowledge ‘‘are...not only complexes of truths but complex truths.’’ Sciencesare, thus, complexes of truths. On the other hand, to a unity of truths therecorresponds the unity of objectualities in the same discipline. Thus, as Husserlstates,11 ‘‘all singular truths of a science belong objectively together,’’ since theobjectualities about which they are truths belong together. To give a concreteexample, the notions of force, velocity and acceleration are objectually related.Thus, they should be studied by a unique discipline, a dynamics. But thatconnection is different from the connection between, e.g., the second law ofNewtonian mechanics, Force = Mass · Acceleration and other primitive orderived laws of Newtonian mechanics. The objectual connection would remainthe same even if the connection of truths about that objectual domain were

8 Hua XVIII, §62, pp. 230–231.9 Ibid., p. 231.10 Ibid.11 Ibid., pp. 231–232.

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different. It should be clear, however, that although their connection shouldnot be confused with the objectual connection, the statements of Newtonianmechanics—as of any science—at first sight belong together precisely becausethe objectualities with which they are concerned belong together.

In §63 Husserl makes it clear, however, that, since not any combination oftruths forms an objective unity, even if they are truths about the same ob-jectual domain, the unity of a science has to be given by a different sort ofunity, namely, a unity of the foundational nexus. Hence, the unity of theobjectual domain is only a necessary but not a sufficient condition for theunity of a science. ‘‘Scientific knowledge,’’ stresses Husserl,12 ‘‘is [always]knowledge from the foundations.’’ But to know the foundation of something,adds Husserl,13 is to acknowledge the necessity of something to be as it is.Necessity means in this context the same as nomic validity, i.e., validity invirtue of laws. Thus, for Husserl the following are equivalent expressions: (i)to understand that a state of affairs is regulated by laws, (ii) to understand thatits truth is necessarily valid, (iii) to have knowledge of the foundation of astate of affairs, and (iv) to have knowledge of the foundation of its truth.14 Ageneral truth is a foundational explanatory law, on which is based a wholeclass of necessary truths.

Husserl divides truths in individual and general, ‘‘[t]he first [of which]contain (explicitly or implicitly) assertions about real existence of individualsingularities, whereas general truths are free from such assertions and allowthe derivation only of the possible existence of individualities (as basedexclusively on concepts).’’15 Contrary to general truths, individual truths areby their very nature casual and, thus, when we speak about explanation ofindividual truths from the foundations, stresses Husserl,16 what is meant is theestablishment of their necessity under certain circumstances. Thus, Husserlmakes it clear17 that if the nexus between two facts is a nomic one, then theexistence of that nexus on the basis of the laws that regulate the nexus of thesort concerned is determined as necessary, but only under the assumption ofthe corresponding circumstances—what in the more recent literature has beencalled ‘‘initial conditions.’’ It should be clear that Husserl is here concernedwith what later authors have called the deductive-nomological model for theexplanation of facts, e.g., of the fact that a determined object dissolves (orfloats) in a determined liquid substance.

Husserl is also concerned with the foundation of general truths, i.e., oftruths, which also have a nomic character with respect to possible applicationsto facts falling under them. In this case we have to refer to general laws thatgenerate the nomic proposition, which is being founded by means of spe-

12 Ibid., §63, p. 233.13 Ibid.14 See ibid, pp. 233–234.15 Ibid., p. 234.16 Ibid.17 Ibid.

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cialization – not individualization - and deductive inference. Thus, as Husserlstates18, the foundation of general laws takes us to some general laws that bytheir very nature cannot be founded and are, thus, called ‘‘fundamental laws.’’Hence, the systematic unity of the ideally closed totality of laws, on which arebased ultimately all laws deductively obtained from them, forms the theo-retical unity. It should be clear that we have here a very general scheme ofdeductive foundation of laws, which applies both to the deduction of lawsfrom axioms in the logico-mathematical sciences and to the explanation oflaws in the physical sciences. The difference between these two sorts of the-ories lies in the foundational laws. In the physical sciences those laws arehypotheses cum fundamento in re, as already explained in Chapter IV of theProlegomena,19 but not explicitly articulated in this last chapter—which wasalready conceptually completed before the remaining chapters were written.20

Moreover, after mentioning that arithmetic, geometry, analytical mechanicsand mathematical astronomy are especially good examples of explanatorytheories, and underlining that the possibility of assuming an explicativefunction is an obvious consequence of the definition, Husserl acknowledges21

that one can understand under theory in a more imprecise sense any deductivesystem, even though its ultimate foundations could very well not be founda-tional in the strict sense, but only preliminary foundations which bring usnearer to the ultimate foundations. This distinction brings to the fore anotherrelated distinction, namely, between explanatory nexus and deductive nexus.As Husserl points out,22 an explanatory nexus is always a deductive one, but adeductive nexus is not always an explanatory one, i.e., one based on funda-mental laws. In the same way, adds Husserl,23 ‘‘[a]ll foundations are pre-misses, but not all premises are foundations.’’ Moreover, all deductions arenecessary, since they are governed by laws. However, although all conclusionsfollow from inferential laws—what we now call inference rules—that does notmean that they follow from laws as premisses and that they are based on suchlaws in the strict sense.24 As Husserl had already stressed25 in his refutation ofpsychologism, to be derived according to logical inference rules should not beconfused with being derived from logical laws as premisses.

Husserl begins §64 with a distinction between essential and extra-essentialunificatory principles. The truths of a science, says Husserl,26 are essentiallyone if their connection is based on what makes a science, namely, knowledge

18 Ibid.19 LU I, §23, p. 83.20 It dates essentially from 1894, whereas the remaining ten chapters of the Prolegomena datefrom 1895. See on this issue Husserl’s Introduction to the Logical Investigations, pp. 35–36.21 Ibid., p. 235.22 Ibid.23 Ibid.24 See ibid.25 LU I §19.26 Ibid. §64, pp. 235–236.

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from the foundations, i.e., explanatory or foundational knowledge in the strictsense. ‘‘Essential unity of the truths of a science is [always] explanatoryunity.’’27 But, as Husserl adds,28 explanatory unity means theoretical unity,homogeneous unity of nomic foundation, i.e., homogeneous unity of explan-atory principles. Sciences in which the unity of principles determines theobjectual region and, thus, embraces in ideal unity all possible facts andgeneral singularities having their explanatory principles in fundamental laws,are called by Husserl29 ‘‘nomological sciences,’’ since their most essentialunifying principle is a nomic one (not an objectual one), or ‘‘explanatorysciences,’’ in case you want to emphazise the explanatory character of theirunity. On the other hand, one can bring together the truths of a science, whichmerely forms an objectual (or ontological) unity. Thus, one connects thetruths that concern, as Husserl liked to say, the same empirical genus. Suchsciences can be called ‘‘concrete sciences’’ or ‘‘ontological sciences.’’ AsHusserl mentions,30 examples of this second group of sciences are history,geography and natural history. Since the explanation by principles in thesesciences could take us to very different and heterogeneous theoretical sci-ences, Husserl says31 that the unity of such sciences is extra-essential. Husserlunderlines32 that the abstract or nomological sciences are the fundamentalones, from which the concrete sciences have to extract all what makes them ascience, i.e., everything theoretical. Of course, remarks Husserl,33 for theconcrete sciences it is sufficient to adhere the objectual with which they areconcerned to the lowest laws of the nomological sciences and indicate in verygeneral terms the route of their possible explanation, since the reduction toprinciples and the construction of explanatory theories is the concern only ofthe nomological sciences. Although Husserl very well knows that the theo-retical interest is not the only decisive one, he makes it clear34 that where thetheoretical interest prevails, the individual and the empirical connection are ofno value, or only as a mere reference point for the construction of the generaltheory. Thus, adds Husserl,35 for a theoretical physicist, e.g., stars are simplyexamples of gravitational masses.

At the end of §64 Husserl considers a second sort of extra-essential prin-ciple of scientific unity, namely, the fundamental norm or value that gives itsunity to a normative discipline. In all such disciplines, it is the fundamentalnorm, which brings together the truths of the discipline, and constitutes theunity of the region. But, as already shown in Chapter II of the Prolegomena,

27 Ibid., p. 236.28 Ibid.29 Ibid.30 Ibid.31 Ibid., pp. 236–237.32 Ibid., p. 237.33 Ibid.34 Ibid., p. 238.35 Ibid.

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normative disciplines are dependent on theoretical ones, especially, on thetheoretical in the strict sense of nomological, since the former extract from thelatter all what is essential for them to be a science.

In §65 Husserl considers conditions of possibility of any science in generalor, more precisely, of theory in general, since scientific knowledge can only beobtained by means of a theory in the strict sense of a nomological science.Moreover, Husserl remarks36 that since theories consist of truths deductivelyconnected, the conditions of possibility of truths in general and of thedeductive unity in general reduce to the conditions of possibility of theories.Husserl is clearly conscious37 of the affinity of the problem under consider-ation with Kant’s problem of the conditions of possibility of experience, andcorrectly considers it a generalization of the Kantian problem.

First of all, Husserl acknowledges38 the existence of subjective conditions ofpossibility of a theory, which are really conditions of possibility of theoreticalknowledge in general, or even of knowledge in general for any human being.Such subjective conditions can be real or ideal. Real conditions, e.g., causalconditions of our thought, do not interest Husserl at all. Ideal conditions canbe noetic, i.e., based a priori on the ideal of knowledge as such, withoutconsidering any empirical aspect of human knowledge; or logical, whichHusserl describes somewhat misleadingly as based exclusively on the contentof knowledge. With respect to the noetic—or, more explicitly, idealsubjective—conditions, it is clear that the thinking subjects should have thecapacity to apprehend the truth of propositions and the relation of deductivefoundation between truths and, in particular, between truths or laws and moregeneral laws that provide their explanatory foundation. With respect to thelogical (or ideal objective) conditions—which are Husserl’s main concernhere—it is clear that truths, and especially, laws, foundations and principlesare what they are with total independence of their being apprehended byanyone. It is not the case that they are valid because we can have evidence ofthem, but we can have evidence of them because they are valid. Thus, asHusserl stresses, the a priori laws ‘‘that belong to truth as such, to a deductionas such, to a theory as such,...express ideal conditions of possibility ofknowledge in general, respectively, of deductive knowledge and of theoreticalknowledge in general, and such conditions are exclusively based on the con-tent of knowledge.’’39 These a priori conditions of knowledge are by no meanssubjective but objective conditions of the possibility of knowledge, since theycan be considered with complete independence of any thinking subject. Intheir meaning content such laws are not concerned with knowledge, judgmentor inference, but with truth, proposition and consequence, i.e., with theobjective content independent of any activity of knowledge of a thinkingsubject.

36 Ibid. §65, p. 239.37 Ibid.38 Ibid.39 Ibid., p. 240.

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This important distinction between ideal noetic conditions of possiblity ofthe knower and ideal logical conditions of possibility of truth and theories,which are completely independent of any knowing subject, and the recogni-tion of the importance of both, have puzzled interpreters of Husserl. On theone hand, he acknowledges the legitimacy of transcendental investigations ofthe ideal conditions of the knowing subject, with which a great part of his laterresearch is concerned, and which has some affinities with the Kantian and neo-Kantian program. On the other hand, Husserl clearly propounds a Platonismof truth and meaning, with clear affinities to Bolzano and Frege. As we willsee below, his views on mathematics are also those of (a sort of) Platonism,though a more refined one than Frege’s and, contrary to the latter’s, a nonreductionist Platonism.40

At the beginning of §66 Husserl states41 that the question about the idealconditions of possibility of knowledge brings to the fore laws based on cate-gorial concepts totally alien to any knowing subject. It is precisely these lawsand the categorial concepts on which they are based that constitutes theobjective-ideal conditions of possibility of a theory in general, where theory isunderstood—as well as truth and law—as an ideal content of possibleknowledge. Thus, for Husserl42 truth is an ideal identical content corre-sponding to a multiplicity of acts of knowledge, whereas a theory is also anideal identical content, although a complex one formed from purely idealblocks, e.g., truths connected by the relation of foundation and consequence,and which corresponds to a multiplicity of complexes of individual acts ofknowledge. In this objective sense the conditions of possibility of a theorywhatsoever concern the possibility of the objects conceptually conceived and,thus, the possible validity or substantiality, as Husserl says,43 of the conceptunder which the object falls. Moreover, Husserl adds44 that the evidentknowledge of any determined theory guarantees the objectual possibility or, inHusserl’s terminology, the substantiality of a theory in general, i.e., thenon-voidness of its concept.

40 There has been a decades long discussion on whether after his transcendental turn of 1907Husserl abandoned the Platonism of LU, and whether his transcedental phenomenology is morecompatible with intuitionism or some sort of constructivism than with Platonism. Certainly, thesuperficial similarity with Kantian views supports such an interpretation, and constructivism notonly is not incompatible with the usual rendering of Husserl’s transcendental phenomenology, butat first sight seems to be the natural philosophy of mathematics of a phenomenologist. None-theless, as a matter of fact, in his writings on mathematics and logic—aside from possible isolatedremarks as the one referred to in the Appendix below—after the transcendental turn Husserlpropounded essentially the same views on an objective but not ontological logic and a formalontological Platonist mathematics as in LU. Moreover, Husserl never seemed to have retracted ofhis classification of Kant’s views in Chapter VII of LU I as a sort of specific relativism. Hence, oneshould not press too much the affinities between Husserl’s and Kant’s transcendental philosophies,and beware of assessing Husserl’s views as a foundation of Brouwer’s Fichtean mathematicalsubject.41 Ibid., §66, p. 241.42 See ibid., p. 242.43 Ibid.44 Ibid, pp. 242–243.

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Husserl is interested in determining the primitive concepts on which theconcept of a theory in general is founded. Moreover, he wants to discover thepure laws, based on such primitive concepts, which confer unity to any theory,i.e., the laws belonging to any theory as such, and which determine a priori thepossible variations or sorts of theories. Husserl stresses,45 that since suchconcepts and ideal laws constitute the idea of a theory in general and, thus,delimit its possibility, any presumed theory can be a theory only if it can besubsumed under such concepts and ideal laws. Thus, as Husserl points out,46

to logically justify a given theory one has to refer it to those concepts and lawswhich ideally constitute any theory whatsoever, and which deductively and apriori regulate any specialization of the idea of theory in its possible types.Hence, Husserl will investigate the a priori theoretical and nomological sci-ence concerned with the ideal essence of science in general, i.e., he will studythe theory of all theories, the science of all sciences.

Meaning categories and formation rules

The first task of Husserl’s theory of all theories is the clarification of theprimitive concepts that make possible a theoretical nexus, i.e., the conceptsconstitutive of the theoretical unity and those which are nomologically con-nected with them. In some sense, Husserl says, we have to deal here withsecond order concepts, i.e., ‘‘...concepts of concepts and of other ideal uni-ties.’’47 Moreover, Husserl stresses48 that theories are deductive connectionsof propositions, which are themselves nothing else than connections of adetermined form between concepts. As Husserl puts it in his extremelyvaluable but relatively recently published Einleitung in die Logik undErkenntnistheorie,49 the whole theoretical content of science is from top tobottom meanings. But sciences and theories in general are structured inpropositions, which are closed unities of meaning. Thus, Husserl adds50 thatby replacing the constitutive parts of a theory by indeterminates we obtain theform of the theory. In this way are obtained also the concepts of concept,proposition and truth. Moreover, to this group belong the concepts ofthe elementary forms of connection so important for the deductive unityof propositions, e.g., the conjunctive, disjunctive and hypothetical connectionsof propositions, which play such a decisive role in the formation of new

45 Ibid., p. 243.46 Ibid.47 Ibid., §67, p. 244.48 Ibid., pp. 244–245.49 Einleitung in die Logik und Erkenntnistheorie, - from now on ELE -, p. 70, Husserliana XXIV,1984.50 LU I, p. 245.

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propositions. Included here are also the forms of glueing together the mostbasic unities of meaning to form elementary propositions, which, as Husserlstates,51 brings us to the different forms of subject and predicate, to the formsof conjunctive and disjunctive binding at the sub-propositional level, and evento the forms of plurals. Husserl underscores52 that there are fixed laws gov-erning the complications formed iteratively, by means of which an infinitemultiplicity of new forms is generated from the primitive forms. Hence, on thebasis of a finite and meager input, by means of these laws (or rules) for theformation of propositions, a whole overview of the infinite multiplicity offorms of propositions is possible.

Both in the Fourth Logical Investigation of the second volume of LogischeUntersuchungen53 as well as in Einleitung in die Logik und Ekenntnistheorie54

(and elsewhere55) Husserl makes explicit the relation between this first levelof logic, which studies the forms of meaning that belong a priori to proposi-tions independently of truth and falsity, and a sort of logico-grammaticalnucleus of natural languages. Thus, this morphology of meanings is also calledby Husserl in both works ‘‘pure grammar,’’ and constitutes, according toHusserl, a common nucleus of logico-linguistic universals present in all lan-guages, in contrast to their multiple empirical diversity not only with respectto the actual vocabulary but with respect to many grammatical forms andrules. As Husserl puts it in Einleitung in die Logik und Erkenntnistheorie,56

the elements of meaning can only be ordered and brought together in somedetermined ways so as to produce a complex unitary meaning. Thus, in thesphere of meanings there are laws which regulate the composition of meaningsand in this way separate sense from nonsense, and which are presupposed byany consideration of the truth or falsity of propositions.

On the other hand, Husserl remarks57 that in nomic connection with theaforementioned primitive concepts, the meaning categories, there are othercorrelative concepts, like those of set, state of affairs, plurality, cardinalnumber, ordinal number, relation, connection, part and whole. (As stressed inEinleitung in die Logik und Erkenntnistheorie,58 the notion of whole is dif-ferent from that of set and irreducible to it.) These concepts are the objectualformal categories, or, better, formal ontological categories since they are theprimitive concepts of mathematics conceived as formal ontology59 and arebased on the notion of object. In both cases we have concepts which are totally

51 Ibid.52 Ibid.53 LU II, U. IV.54 ELE, pp. 71–72.55 See, e.g., Formale und transzendentale Logik—from now on FTL—, 1929, Husserliana XVII,1974, as well as Logik und allgemeine Wissenschaftstheorie, Husserliana XXX, 1996.56 ELE, pp. 72–73.57 LU I, p. 245.58 ELE, p. 78. See also FTL, p. 82.59 See on this point FTL, §§23–27.

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independent of any particular knowledge content, and under which have to besubsumed all concepts, propositions, objects, states of affairs, etc. present inany thought activity. Thus, it is a primary task of logic to fix all such meaningand formal ontological categories, and clarify their nature.

It should be perfectly clear that what Husserl has in mind when consideringthe meaning categories and the laws of their iterative complications to formcomplex propositions of any finite degree of complexity is nothing else thanwhat Carnap three and a half decades later—without mentioningHusserl—called ‘‘rules of formation’’60 and is now current stuff in rigorouslogic books. But Husserl’s scope is larger, since it embraces also subproposi-tional meanings, and points to a somewhat parallel treatment of naturallanguage. This will become much clearer in the Fourth Logical Investigation,where he develops the notion of a pure logical grammar based on similarnotions but applied to natural language. Thus, Husserl is also a forerunner ofthe categorial grammar current nowadays. It is appropriate to call this firstlevel of Husserl’s theory of theories the logico-grammatical level.

Logic, mathematics and the mathesis universalis

The second group of problems considered by Husserl concerns the laws basedon those two sorts of categorial concepts which, instead of studying the pos-sible forms of complexions and modification of theoretical units, studies theobjectual validity of the forms thus originated, and on the objectual side, theexistence or non-existence of the objects in general, the states of affairs, sets,numbers etc.61 These laws which deal with meanings and objects in the mostgeneral way possible—the logico-categorial—constitute theories. Thus, wehave, on the side of meanings, the different theories of inference, of which thetraditional syllogistic is just an example among others, and on the objectualside, we have, e.g., number theory, which is based on the formal-ontologicalcategory of number. The whole group of laws belonging to the different for-mal-syntactical and formal-ontological theories are based on a small group ofprimitive (or fundamental) laws which have their origin in the categorialconcepts.

On this issue, it should be stated, firstly, that on the logical side Husserl isdealing here with the logical laws of propositional and predicate logic, and, ingeneral, with all possible logical laws, which contrary to the laws of the firstlevel, protect against formal countersense, not against nonsense. In Einleitungin die Logik und Erkenntnistheorie62 Husserl makes it clear that propositional

60 Die Logische Syntax der Sprache, 1934, expanded English version, 1937, reprint 2003, §1, p. 2,§2, p. 4.61 See LU I, §68, p. 247.62 ELE, pp. 435–436.

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and predicate logic are the most basic of these theories, although he alsoremarks63 that the complete development of the most basic theories willprobably require the use of inference forms from less basic theories ofinference. Hence, the constitution of these theories is not simply hierarchical.

On the other hand, Husserl makes it clear that formal ontology includes allformal mathematics, i.e., all mathematics with the exception of physicalgeometry. Thus, as Husserl states in Einleitung in die Logik und Erkenntnis-theorie,64 mathematics includes among others the pure doctrine of cardinalnumbers and that of ordinal numbers, combinatorics, all the disciplines ofmathematical analysis, including, e.g., the theory of functions, number theory,algebra, and both the doctrine of Euclidean multiplicities and that of non-Euclidean multiplicities in general, thus, as he observes,65 the whole realm ofwhat Felix Klein had called ‘‘arithmetical mathematics.’’ Moreover, as men-tioned above,66 even a (still not developed) theory of parts and wholes, amereology, should be included in this broad notion of formal-ontologicalmathematics.

In view of the homogeneity of these categorial concepts an all embracingtheory is constituted of which the aforementioned theories are relativelyclosed components. In virtue of the formal generality of this all embracingtheory of all possible meanings and all possible objects, each and every sci-entific theory has to be subsumed under it in order to be valid. As Husserlstresses,67 this does not mean that each singular theory presupposes each ofthese laws as foundation of its possibility and validity. It is simply that thoseformal theories and categorial laws form the common all embracing ground,‘‘of which each particular valid theory extracts the ideal foundations of itssubstantiality in virtue of its form.’’68 The validity of a theory can be estab-lished only in virtue of its form and on the basis of this all-embracing groundas an ultimate foundation. Moreover, as Husserl stresses,69 since valid theoriesare complex unities of interconnected truths, it is clear that the laws thatconcern both the concept of truth and the possibility of singular deductiveconnections of such and such forms should be included here.

On this last point, there is some unclarity in the exposition of the Prole-gomena, not only because the intention is syntactic but the vocabularysemantic, but also because the concept of truth, which is clearly semantic, isthrown into the mix. In Formale und transzendentale Logik70 Husserl makes itclear that this second level of the logico-mathematical building is, on thelogical side, a purely syntactical one. It is the study of theories of inference and

63 Ibid., pp. 436–437.64 Ibid., pp. 70–71.65 Ibid. p. 55.66 See above, footnote 39.67 See on this issue, e.g., ELE, pp. 59–60 and LU I, §68, pp. 247–248.68 See ELE, p. 61 and LU I, §68, p. 248.69 LU I, §68, p. 248.70 FTL, p. 70.

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derivability from fundamental laws, the level of deducibility and non-contradiction, without any concern for truth or other related concepts. It is thelevel much later called by Carnap71—once more, without any reference toHusserl—the level of logical syntax, i.e., the level of transformation rules.

Husserl, however, did not ignore the semantic side of logic. Precisely inFormale und transzendentale Logik72 Husserl adds a new stratum to hisconception of logic, namely, a logic of truth built immediately on the syn-tactical theory of deduction. This stratum would be obtained from the stratumof logical deduction by the introduction of the concept of truth and relatedsemantical concepts, together with the laws that govern them. Thus, eventhough—contrary to the first two strata of logic—Husserl said very little aboutthe logic of truth, also on this point is he a forerunner of later developments byTarski, Carnap and others. Moreover, as Husserl underscores,73 when we takeinto account the semantic side, the ‘‘logic of truth,’’ it becomes clear that alllogical statements presuppose a region of individuals, a world of individuals, inwhich they are valid. Thus, Husserl’s conception of logic, no matter howformal and devoid of any content, is not that of a so called free logic, but tendsto coincide also in this sense with our now classical mathematical logic.74

The theory of all theories

As Husserl mentions,75 the former level of research was sufficient to fix theconditions of possibility of any theory in general. But this points to a stillhigher level of study, namely, to the a priori investigation of all forms—orsorts—of theories and the corresponding laws governing their relations. Thusoriginates the possibility of a still more embracing theory, which studies theconcepts and essential laws constitutive of the concept of theory, and whichproceeds to differentiate this notion, and study, not the possibility of a par-ticular given theory—as in the former level—but a priori the possible theories.Hence, on the basis of the former level’s investigation, we are now able toconsider the multiple pure forms of theories, whose substantiality, i.e.,objectual possibility, has already been established. An especially decisive as-pect of this new level is the study of the relations between forms of theories. Itwill be possible not only to obtain the possible forms of theories, but to havean overview of their nomological nexus and, thus, of the possible transfor-mations of one of these forms into another through variation of some

71 See footnote 40 above.72 FTL, pp. 60–61, 70–71.73 Ibid. pp. 212–213.74 It should be pointed out, however, that in Alte und neue Logik: Vorlesungen 1908/09, pp. 230–232, Husserl extends his notion of logic in order to include the treatment of logical modalities,thus, of modal logic, and both therein—p. 230—and already in LU I—see §10 below—to includethe theory of probability.75 LU I, §69, p. 248.

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fundamental aspects. Moreover, adds Husserl,76 if not in general, at least forthe theory of forms of some determined sorts or, in Husserl’s terminology,genres there will be general propositions that will govern not only the separatedevelopment of the forms of theories, but especially the connection betweenthe forms of theories and their transformations, according to laws, into others.As Husserl clearly underlines,77 the propositions with which we are hereconcerned are of a different nature than the axioms and theorems of theformer level (e.g., of syllogistics or arithmetic). They are clearly of a veryabstract metamathematical level, and their deduction can only be based on thenature of such forms of theories, since at this level there are no axiomsproperly.

Furthermore, Husserl stresses78 that the subsumption of a theory under itsform, as done in the former level, has a great methodological importance,since the expansion of the deductive and theoretical realm enhances thevitality of the theoretical research and contributes to the richness andfecundity of its methods. Thus, adds Husserl,79 the solution of problemsoriginated in a given discipline can, under determined circumstances, receivesome methodological help by a return to its category type, i.e., to the form ofthe theory, and from this eventually through a transition, even to a moreembracing form and its laws.

Finally, it should be briefly mentioned that in Formale und transzendentaleLogik80 and elsewhere Husserl related his theory of all theories with Hilbert’sphilosophy of mathematics. Specifically, Husserl went so far as to claim a sortof completeness of the theory of all theories, which seemed to conflatedeductive (or syntactic) completeness with semantic completeness.81 Justrecently the interpretation of Husserl’s views on completeness has been thefocus of some important contributions.82 However, it would take us too far toadequately discuss here this interesting issue, especially since a revised ver-sion of Husserl’s lectures on completeness has been recently published byElisabeth Schuhmann and the late Karl Schuhmann.83

76 Ibid., p. 249.77 Ibid.78 Ibid.79 Ibid.80 FTL, pp. 98–102. See also Appendixes VI, VII, VIII and X of the Husserliana edition ofPhilosophie der Arithmetik.81 In FTL, p. 100 there are two formulations of the completeness requirement, which seem to pointto what we now call semantic and deductive completeness. In any case, these two notions were notclearly differentiated until the epoch-making papers of Godel and Tarski mentioned in the ref-erences.82 See on this issue the interesting paper by Ulrich Majer included in the references, and especiallythe detailed analyses by Jairo J. da Silva in ‘‘Husserl’s Two Notions of Completeness’’ and in ‘‘TheMany Senses of Completeness,’’ which reject the rendering of the conflation of semantic anddeductive completeness.83 See Elisabeth Schumann and Karl Scuhmann’s ‘Husserls Manuskripte zu seinem GottingerDoppelvortrag von 1901’.

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Husserl’s theory of manifolds

At the beginning of §70 Husserl states84 that the mathematical theory ofmanifolds current in his day—that of the tradition of Riemann, Helmoltz,Klein, Lie and others—is a correlative partial realization of his ideal of atheory of (deductive) theories, even though mathematicians have not clearlygrasped the nature of this new discipline and have not risen to the highestabstraction of an all embracing theory. Thus, as Husserl underlines: ‘‘Theobjectual correlate of the notion of a determined possible theory in virtue ofits form is the notion of a region of possible knowledge to be governed bymeans of a theory of such a form.’’85 Following the mathematicians’ usage,Husserl calls such a region a ‘‘manifold.’’ On this point, it should be clear fromHusserl’s remarks in Einleitung in die Logik und Erkenntnistheorie86 that forhim a manifold is a collection or class of objects thought in complete inde-termination and generality, together with some connections between the ob-jects for which some given laws are valid. Moreover, in LogischeUntersuchungen he stresses,87 that the region ‘‘...is uniquely and exclusivelydetermined by being under a theory of that form, i.e., by being possible for itsobjects some connections which obey certain fundamental laws of such andsuch determined form,’’ which is what is decisive here, since the objects arecompletely indeterminate with respect to their material nature. They are, saysHusserl, ‘‘neither directly determined as individual or specific singularities,nor indirectly by means of their material types or genres, but only by means ofthe forms of their respective connections.’’88 Moreover, adds Husserl89 ‘‘[suchconnections] are as indeterminate in their content as their objects; [since] onlytheir form is determined by means of the forms of the basic laws assumed tobe valid for them.’’ Moreover, these laws determine both the form of theregion and the form of the theory to be built. As an example, Husserl men-tions90 that in the doctrine of manifolds the sign ‘+’ is not the addition sign fornumbers, but the sign for any connection for which are valid laws of the form‘a + b = b + a’. The ‘‘conceptual objects’’ of the multiplicity make possiblethose fundamental operations and others compatible with them, and in thisway completely determine the manifold. This last point is more explicitlyexpressed in Einleitung in die Logik und Erkenntnistheorie91 by consideringthree possible interpretations of ‘a + b = b + a’, namely, the arithmetical one,the set-theoretic one with ‘+’ rendered as the set-theoretic union, and thegeometrical one with ‘+’ rendered as the juxtaposition of straight lines. More

84 LU I, §70, p. 250.85 Ibid.86 ELE, p. 88.87 LU I, p. 250.88 Ibid.89 Ibid.90 Ibid., p. 251. See also FTL, p. 105.91 ELE, p. 85.

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generally, says Husserl,92 if you have, e.g., a collection of number-theoreticaxioms, and ‘‘...in a completely different region [of mathematics] a collectionof axioms is valid, which completely coincides in its form with the collection ofarithmetical principles, ...then...to every number-theoretical proposition therecorresponds a proposition in the new region and viceversa,’’ and thus, not onlythe principles but also their consequences, inferences, proofs and theorieshave the same form, i.e., they are equiform. Thus, it is not necessary to makethe deduction twice. As soon as one acknowledges the equiformity of theprinciples, one knows a priori that everything has also to be equiform. In thisway, as Husserl adds,93 one can emancipate the form of a mathematical theoryfrom the objects governed by the theory. Thus, the mathematical theory be-comes the theory of an indeterminate region of objects for which the laws ofthe theory are valid. It is unnecessary to stress here how mathematics in thetwentieth century developed in the direction pointed at in 1900 by Husserl.Universal algebra and general topology are clearly partial realizations of theHusserlian ideal.

As Husserl puts it: ‘‘The most general notion of a doctrine of manifolds isthat of a science which definitely forms the essential types of possible theories(respectively, regions) and studies the nomic relations between them.’’94

Hence, as Husserl underlines,95 any actual theory is really a specialization orsingularization of its corresponding form of theory, and any theoretical regionof knowledge is simply a singular multiplicity. Thus, if in the theory of man-ifolds one develops completely the formal theory concerned, then one hascompleted all theoretical work for the structuring of the actual theories of thesame form. In Einleitung in die Logik und Erkenntnistheorie,96 Husserl con-siders more explicitly the possibility of combining in a system different the-ories to obtain a complex but compatible manifold. More generally, beginningwith the formal type of a mathematical structure, one can modify the forms soas to make it possible to combine and connect by means of laws differentpossible multiplicities. It should be clear that Husserl once more is antic-ipating future developments in mathematics, namely, the possibility of com-bining different but compatible mathematical structures to obtain a complexmathematical manifold. Topological groups are a beautiful example of whatHusserl has here in mind.

Husserl stresses97 that it is not possible to really understand the mathe-matical method without considering the doctrine of manifolds, and the sub-sumtion of theories under their most embracing forms. As an example of thispoint of view, Husserl mentions Riemann’s doctrine of manifolds, which is a

92 Ibid.93 Ibid.94 See LU I, p. 251.95 Ibid.96 ELE, pp. 86–87. See also ‘Husserls Manuskripte zu seinem Gottinger Doppelvortrag von 1901’,p. 91.97 LU, p. 251.

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generalization of geometrical theory. More generally, when Husserl speaks ofa generalization of geometrical theory he refers to Riemann’s doctrine ofn-dimensional manifolds, whether the manifold is Euclidean or not, as well asto Grassmann’s and others’ related theories. On this point, Husserl refers alsoto Lie’s doctrine of transformation groups and to the investigations of hisfriend Georg Cantor.98 It is, however, Riemann’s pioneering notion of man-ifold which seems to have exerted the greatest influence on Husserl’s views onmathematics. It seems appropriate to say a few words about Riemann’s rev-olutionary notion of a mathematical manifold.

On Riemann’s notion of a manifold

At the very beginning of his duly famous inaugural lecture Uber dieHypothesen, welche der Geometrie zugrunde liegen, Riemann observes99 thatgeometry has traditionally taken the concept of space and the most basicconcepts for the constructions in space as given. Moreover, he adds100 thatneither mathematicians nor philosophers were capable of eliminating theobscurities related to such concepts because they lacked the general conceptof a multiply extended magnitude, under which spatial magnitudes are to besubsumed. Riemann’s main task is to obtain the concept of a multiplyextended magnitude from general concepts of magnitude. An importantconsequence of this procedure is that a multiply extended magnitude iscapable of different measuring relations, i.e., measuring (or metrical) relationsare not intrinsic to the notion of a multiply extended magnitude. Physicalspace, i.e., the space of our physical world, is only a special case of a three-foldextended magnitude. Hence, it follows that the (physico)geometrical state-ments are in no way derivable from the general concepts of magnitude, andthat those properties which distinguish physical space from other three-foldextended magnitudes can be obtained only from experience. Thus, the factsfrom which the metrical relations of space are determined are, as any fact, notnecessary, but possess only empirical certainty.

Such assertions of Riemann constitute a really deep conceptual revolution,mathematically and philosophically deeper than the mere discovery of the twomain sorts of non-Euclidean geometries by Gauß, Bolyai and Lobachevsky,on the one hand, by Riemann himself, on the other hand. Riemann’s remarksare an important breakthrough, a rupture with the tradition both of geometerswho, even in the face of non-Euclidean geometry, still saw these new geom-etries as mere conceptual objects, and with the powerful Kantian traditionwhich gave them its philosophical foundation. But there is in Riemann’s

98 See ibid., p. 252. To give an idea of this influence, it should be mentioned that in SAG Husserlmentions Cantor on pp. 24, 40, 82–84, 95, 145, 240, 244 and 413; Grasssmann on pp. 242, 245, 253,256, 391, 396 and 401; Klein on p. 397; Lie on pp. 397 and 412; Helmholtz on p. 160; and Riemannon pp. 95, 250, 256, 323–324, 329, 330, 337–344, 347, 406, 407, 409 and 411–413.99 Uber die Hypothesen, welche der Geometrie zugrunde liegen, p. 1.100 Ibid.

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epoch-making monograph a more general breakthrough with the Kantiandoctrine of space. In the Transcendental Aesthetics of the Kritik der reinenVernunft,101 Kant had argued for the intuive—as against the concep-tual—nature of space that concepts are such that they cannot contain inthemselves, as space does, infinitely many parts of exactly the same nature,e.g., obtained by division of any spatial magnitude. Concepts are, for Kant,such that they can be present, as what Kant called representations, in a finiteor infinite variety of cases, like the concept of ‘horse’ is present in our rep-resentation of any given horse, or the concept of ‘angel’ is present in ourrepresentation of any angel, but they cannot contain in themselves an infinitemultiplicity of parts of the same nature. However, Riemann states in hismonograph102 that concepts of magnitude are possible only when there is ageneral concept which admits of different individuations (or, as he calls them,‘‘modes of determination’’). These individuations are of two sorts, dependingon the nature of the general concept. The individuation can be such that thetransit from an individual to another is a continuous one, as in the case ofspace, or it can be a discrete one, as in the case of horse, angel or naturalnumber. In the first case, the individua constitute a continuous manifold, inthe second case a discrete manifold. In the first case the individua are called‘‘points,’’ in the second they are called ‘‘elements.’’ Clearly demarcated partsof a manifold are called by Riemann quanta, and their comparison is obtained,in the case of discrete manifolds, by means of counting, and in the case ofcontinuous manifolds, by means of measurement.

Here we have a very general philosophical distinction between discrete andcontinuous manifolds, which seems as inclusive as Kant’s distinction betweenconcepts and intuitions, and, for the sake of comparisons, corresponds in somesense to it. (Of course, Kant, who was so limited by traditional logico-philo-sophical distinctions, could not even foresee the scope of Riemann’s use of theterm ‘‘concept.’’) But what is decisive in Riemann’s distinction is not only itsgenerality, but the fact that in both sorts of manifold we are dealing withconcepts. The notion of a continuous magnitude, which in Riemann’s work isthe general scheme on which his conception of a multiply extended manifoldis founded, is not an intuition but a concept. Such a view clearly runs counterto Kant’s argument on behalf of the a priori nature of space (and time)already alluded to above, according to which space (and time also) is primarilyan intuition, not a concept, since the notion of space contains in itself alreadyan infinity of individua such that the individua can be divided indefinitelywithout obtaining anything else than individuals of exactly the same nature,whereas in the case of concepts such an infinite division which would preservethe nature of the notion would be impossible. (Parts of horses are not horses,although the concept of horse is present in a potentially infinite multiplicity ofconcrete horses.) Contrary to the Kantian tradition, Riemann considers thatcontinuous manifolds are of the same conceptual nature as the more common

101 Kritik der reinen Vernunft, B, p. 40.102 Uber die Hypothesen, welche der Geometrie zugrunde liegen, p. 3.

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discrete ones. Thus, since continuous manifolds are present in so many partsof mathematics—whereas discrete ones in a very general form are present inits remaining parts—, there is no place for intuition in mathematics. Puremathematics is a purely conceptual discipline. Hence, it is no mystery that, asstated at the very beginning of the monograph103 and repeated many timesthroughout it, the nature of physical space has to be determined by empiricalmeans. Moreover, it should be mentioned that Riemann also asserts104 that forthe doctrine of extended manifolds nothing is presupposed which is not al-ready contained in the concepts. Thus, according to a popular definition ofanalyticity, which comes from Kant and even from Leibniz, the doctrine ofextended manifolds deals exclusively with purely analytic statements. Hence,the blow on the Kantian tradition is also n-fold.

It is unnecessary to discuss further Riemann’s extraordinary monograph. Itsdecisive importance for the development of the theory of functions, itsgroundwork for the development of topology and of differential geometry arewell documented, as is also the importance of his views for physics, which havebeen fully appreciated only after the development of general relativity.105

Because of his mathematical formation under Karl Weierstrass and hisphilosophical genius, Husserl was one of the philosophers—if not the phi-losopher—who better appreciated the importance of Riemann’s notion ofmanifold for mathematics, as well as the contributions of others, especiallyKlein and Lie, and developed a view of the nature of mathematics which insome sense anticipated by almost half a century the views of the Bourbakischool.106 For the sake of completeness, a few words should be said about theBourbakian view of mathematics.

A brief note on Bourbaki’s view of mathematics

The school of mostly French mathematicians under the ficticious name ofNicholas Bourbaki developed since the late 1930’s a whole systematization of

103 Ibid., p. 1.104 Ibid., p. 4.105 For a much more detailed treatment of Riemann’s views, there are two excellent works,namely, the very valuable book by Erhard Scholz Geschichte des Mannigfaltigkeitsbegriffs vonRiemann bis Poincare and Roberto Torretti’s encyclopedic Philosophy of Geometry fromRiemann to Poincare.106 It should be pointed out here that although Husserl rejected since about 1890 Kant’s views onthe synthetic a priori nature of three dimensional Euclidean space, he considered, however, thatthere are some synthetic a priori features of intuitive space. On this point, he clearly differed fromRiemann, whose views did not leave room for any intuitive space. For a beautiful treatment ofspace, including a first approach to Husserl’s views and their relation to the Riemann-Einsteinianconception, see Rudolf Carnap’s dissertation Der Raum. Nonetheless, Husserl’s views on theapriority of space are more sophisticated than Carnap’s, since it is not limited to the topologicalproperties but also includes the affine (and projective properties), as well as some basic metricproperties related to the notion of congruence. See on this issue the present author’s forthcomingreview of Husserl’s Alte und neue Logik; Vorlesungen 1908/09.

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mathematics, which has been considered paradigmatic by twentieth centurymathematicians. Based on the tradition of set-theoretical foundations domi-nant in most of that century, they offered a view of mathematics as a theory ofstructures. For (the collective mathematician) Nicholas Bourbaki, mathe-matics is in some sense a hierarchy of structures.107 First of all, there are atleast three basic sorts of (set-theoretic) mathematical structures, which hecalls108 mother-structures, namely, algebraic structures, which are determinedby at least one law of composition,109 order structures and topological struc-tures. Each of these sorts of structures includes a great diversity of mathe-matical structures. Thus, one has to distinguish between the most generalstructure of each sort and the remaining structures of the sort, obtained by theaddition of new axioms,110 e.g., an order structure can be enriched to obtain alinearly ordered structure, by adding an axiom that states that all members ofthe universe of the structure are comparable, or well-ordered structures, byadding that each non-empty set of members of the universe has a firstmember.111 As clearly shown by way of examples somewhat similar to thatgiven by Husserl in Einleitung in die Logik und Erkenntnistheorie mentionedabove,112 very different mathematical structures fall under the same axi-oms.113 Apart from those structures that are exclusively algebraic, or exclu-sively order or exclusively topological structures, there are many mixed (ormultiple) structures, which contain two or more of the mother-structurescombined by one or more axioms which connect the different componentstructures.114 Of course, topological algebra is a clear example of an area ofmathematics in which such mixed structures are present. Thus, as Bourbakiclearly states,115 in this area there are one or more (algebraic) laws of com-position together with a topology and connected by the condition that thealgebraic operations be continuous functions with respect to the topologyunder consideration. Moreover, not two but the three sorts of mathematicalstructures can be present in a mixed structure. As a matter of fact, the morefamiliar and concrete structures, like the structure of the real numbers, aremixed structures. Of course, in the case of these familiar structures we are notconsidering anymore abstract structures, in which the members of the uni-

107 See ‘‘The Architecture of Mathematics,’’ p. 228.108 Ibid.109 See ibid., p. 226.110 See ibid., p. 228.111 Of course, if you add the axiom for well-ordered sets to the order axioms, you obtain thecomparability of all members of the structure by considering all sets of pairs and, thus, do not needany comparability axiom.112 See above, footnote 66.113 See ‘‘The Architecture of Mathematics,’’ pp. 224–225, 226–227.114 See ibid., p. 229.115 Ibid.

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verses under consideration are indeterminate, but more particular entitieswith, as Bourbaki says, ‘‘a more definitely characterized individuality.’’116

As Bourbaki underscores in a footnote,117 in a strict sense, we should haveto consider higher sorts of structures in what could be called a hierarchy oftypes, namely, structures in which the relations do not apply to the membersof the universe but to subsets of the universe or even to sets of sets of highertype. For our purpose, however, it is sufficient with what has been said.118 Thefact of the matter is that Bourbaki’s conception of mathematics is very similarto that of Husserl. The main difference lies in the priviledged position occu-pied by the notion of set in Bourbaki’s conception, a status that has beenquestioned in the last decades by the category theorists. The mother-struc-tures play in Bourbaki’s conception the same role played in Husserl’s by themost general structures based on the formal ontological categories. Thus, settheory, number theory, mereology and others would play the role of themother-structures in Husserl’s conception. There are at least three reasonsthat could be used against the priviledged status of set theory and, thus, thatfavor Husserl against Bourbaki. The first one is that, as Bourbaki himself hasstated,119 there are results in number theory that resist the classification underany of the known structures. The second argument is that, if you include a sortof mereology as a mathematical discipline, as Husserl wanted, its (non-arti-ficial) reduction to set theory could be a problem. But by far the mostimportant argument is that the notion of set can also be defined both in thetheory of categories, as shown in any textbook of category theory,120 and as aspecial case of the notion of relation.121 Thus, the priviledged status of settheory seems unjustified. What seems more natural is that there are somebasic very abstract mathematical notions, which could somewhat artifi-cially—and due to their abstract nature—be interdefinable, without therebeing one more fundamental than the other, and that such notions are thecornerstones of the whole of mathematics in the way described by Husserl.

Another point of discrepancy between Husserl and Bourbaki concerns therelation between mathematics and logic. For Bourbaki, logic seems to haveonly a subsidiary role as the language of mathematics.122 It seems unnecessary,

116 Ibid.117 Ibid., p. 226.118 For a much more complete exposition of Bourbaki’s conception of mathematics, see Chapter 1and especially Chapter 4 of his Elements des Mathematiques: Theorie des Ensembles.119 ‘‘The Architecture of Mathematics,’’ p. 229.120 See, e.g., Saunders Mac Lane’s Categories for the Working Mathematician.121 See, e.g., Hartry Field’s, Realism, Mathematics and Modality, pp. 20–21, as well as SaundersMac Lane’s Mathematics: Form and Function, pp. 359 and 407. Indeed, as Philippe de Rouilhanreminded me after reading a preliminary version of this paper, already John von Neumannpointed out in 1925 to the possible definition of sets in terms of functions. Functions and relationsare, on the other hand, interdefinable, as shown by combining the set theoretic approach withFrege’s.122 On this issue, see Bourbaki’s ‘‘Foundations of Mathematics for the Working Mathematician,’’pp. 1–2.

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however, to dwell on this issue here, especially since Bourbaki’s comments onlogic are very sparse.

The division of labor: mathematicians and philosophers

The exposition of Husserl’s philosophy of mathematics as presented in Log-ische Untersuchungen and elsewhere has been essentially completed. In theremaining sections of Chapter XI Husserl adds a few ‘‘footnotes’’ to thetheory. In §71, Husserl states123 that to tackle the difficult problems facing thedevelopment of his program a division of labour between mathematicians andphilosophers has to be implemented. Thus, the construction of the theoriesand the rigorous solution of all formal problems remains in the domain ofmathematicians. More specifically, Husserl makes clear his support of thework of mathematicians who have developed new sorts of logical inferenceunknown to traditional logic. Hence, it is not the mathematician who crossesthe frontiers of his own field when he produces such results, but the philos-opher when he dares to reject them, since, as Husserl clearly states,124 themathematical treatment of logic is the only scientific one, and also the onlyone that could bring a completion to the task of searching for logical forms ofinference and give us a global view of all the possible problems and all thepossible forms of solution. Moreover, the development of all theories properlyspeaking is the task of the mathematician. Although Husserl does not mentionany specific name, due to his acquaintance with the works of Frege andSchroder, Husserl seems here to be applauding their logico-mathematicalendeavors and defending them from the critiques of the traditional philoso-pher-logicians of his time. On this issue, it should be mentioned that in Ein-leitung in die Logik und Erkenntnistheorie,125 Husserl makes very similarremarks concerning the mathematical treatment of logic and the blindness ofphilosophers like Windelband and even Frege’s teacher Lotze to appreciate it.He is, however, critical of the mathematical logicians’ attempts to explain thecognitive value and sense of their fundamental concepts and principles. (Onthis point, it should be remembered that Husserl had very sharply criticizedSchroder in his review of the latter’s Vorlesungen uber die Algebra der LogikI126 as well as Frege,127 whom he seems to have considered half mathemati-cian and half philosopher, and, thus, not philosopher enough to ade-quately assess the logico-mathematical developments to which he so muchcontributed.)

123 LU I, § 71, p. 254.124 Ibid.125 ELE, p. 162.126 ‘‘Besprechung von Ernst Schroder, Vorlesungen uber die Algebra der Logik I,’’ 1891, reprintedin Aufsatze und Rezensionen (1890–1910), Husserliana XXII, 1979, pp. 3–43.127 In Philosophie der Arithmetik, Chapters VI-IX.

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In a similar vein, in Logische Untersuchungen Husserl asserts128 that themathematician is only the ingenious technician who develops theories ofnumbers, inferences and manifolds without a clear insight in the nature of atheory in general and in the concepts and laws constitutive of that nature.Hence, the technical development done by the mathematician has to becomplemented by an epistemological reflection, motivated only by theoreticalinterests. Such is the task of the philosopher. It is the philosopher who isworried about the nature of a theory, about what makes possible any theorywhatsoever. The philosophical research builds on the results of the mathe-matician, as well as on those of the natural scientist, and constitutes genuinetheoretical knowledge. As Husserl clearly states,129 the six logical investiga-tions of the second volume will provide a philosophical preparation for thetheory of all theories by elucidating ‘‘what the mathematician does not wantto do and cannot do, but has to be done.’’130

Husserl is somewhat unjust on this point not only to Frege, but to his highlyregarded Riemann and to his friend and also highly regarded mathematicianGeorg Cantor, both of whom had interesting philosophical insights. None-theless, such remarks by Husserl, the mathematician turned philosopher, areof particular importance for the assessment of Husserl’s views on philosophyand its relation to (rigorous) science. Contrary to most of his so-called phe-nomenological followers, Husserl acknowledges and applauds the investiga-tions of mathematicians and natural scientists. But that does not mean thatthere is no proper realm of philosophical theorizing, as contemporary natu-ralists would make us believe. Philosophy is not limited to a task of merelyestablishing deductive relations between statements, as some logical empiri-cists in their heyday would have liked, nor is philosophy supposed to assumeany other subsidiary role with respect to natural science, as the propounders ofthe so-called naturalized epistemology—or better, denaturalized epistemol-ogy—would like. Philosophy, in Husserl’s views, complements mathematicsand natural science with a theoretical investigation into the foundations oftheories and knowledge. Philosophy is foundational research in the mostradical and complete way possible. This view of philosphy and its relation bothto the formal and the material sciences will be maintained by Husserlthroughout his whole philosophical career, independently of the small varia-tions of emphasis. Specifically, it should now be clear that there is no rupturebetween the first and the second volume of Logische Untersuchungen, and thatthe apparent rupture between his opus magnum and his later transcendentalphenomenology is more a change of emphasis than a radical one. It is amethodological radicalization which does not affect Husserl’s view of phi-losophy as foundational par excellence. Interestingly enough, it is Husserl’sfamous (but unacknowledged) disciple Rudolf Carnap, who better assessed

128 LU I, §71, pp. 254–255.129 Ibid., p. 256.130 Ibid.

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the nature of Husserl’s transcendental phenomenological turn: it is a meth-odological not a substantial procedure.131

On empirical science

At the beginning of §72 Husserl writes: ‘‘Since no science is possible withoutan explanation from its foundations, i.e., without a theory, pure logic [con-ceived as a theory of theories] embraces in the most general sense the idealconditions of science in general.’’132 That does not mean that pure logic soconceived contains all the ideal conditions of science in general. Although asany sort of science, the theoretical content of empirical science has to besubsumed under the above mentioned laws of the mathesis universalis,empirical sciences are never reducible to their pure theories. Nonetheless, thisaspect of empirical science, which embraces the somewhat complex process ofknowledge in which empirical sciences originate and are modified in thecourse of scientific progress, also lies, stresses Husserl,133 under ideal laws, notonly under empirical laws. This brings us to probability theory.

As is well known, remarks Husserl,134 any theory in the empirical sciencesis a supposed theory. ‘‘It does not offer an explanation from evidently truelaws, but only from evidently probabilistic foundational laws.’’135 Hence,empirical theories have only evident probability and are, thus, provisional,never definitive. Something similar happens, according to Husserl, with thefacts themselves that are to be theoretically explained. It is also the task ofempirical science to explain facts through laws, i.e., nomologically, fromexplanatory hypotheses, which we accept as probabilistic laws. But in thisprocess, states Husserl,136 the facts do not remain completely unchanged.They are modified in the process of knowledge.

Husserl considers that in the procedures of factual science rules a sort ofideal norm. When, e.g., new empirical data tend to disconfirm a theory ac-cepted with nomic probability, we usually do not infer that the foundation ofthe theory was false, but conclude that the theory was correct on the basis ofthe previous data and is correct no more. On the other hand, adds Husserl,137

we sometimes judge that a theory is not correctly founded, even though it is

131 See Der logische Aufbau der Welt, § 64, p. 86.132 LU I, §72, p. 256.133 Ibid., pp. 256–257.134 Ibid., p. 257.135 Ibid.136 Ibid.137 Ibid., p. 258.

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the only one adequate to the present data. (Maybe Einstein’s, Bohm’s andothers’ later struggles with quantum theory can be seen as an illustration ofthe point brought here by Husserl.) Thus, concludes Husserl,138 even in thesphere of empirical thought, where we are concerned with mere probabilities,there are ideal laws, ‘‘on which is based a priori the possibility of empiricalscience in general [and] of the probabilistic knowledge of reality.’’ Moreover,stresses Husserl,139 ‘‘[such a] sphere of pure nomology ... is not related to theidea of theory and, more generally, to that of truth, but to the empirical unityof explanation, respectively, to the idea of probability,’’ and builds a secondimportant foundation of what Husserl called practical logic, and which couldbetter be called scientific (or logical) methodology.

Without trying to extend us unnecessarily, it should at least be brieflymentioned here that for Husserl the explanation from foundations was anessential feature of all science. In the case of the empirical sciences, we have,on the one hand, the explanation of facts and, on the other hand, the expla-nation of laws from laws of a higher level. As we have mentioned else-where,140 the laws of higher level are called by Husserl ‘‘hypotheses cumfundamento in re,’’ since they are not simple empirical laws obtained throughinductive procedures, but theoretical laws, partially based on experience, butintroduced as hypotheses to serve as explanatory foundations of laws of lowerlevel. Newton’s law of gravitation is an example of this sort of laws. Butexplanation does not need to be a purely deductive one. As mentioned above,in the case of empirical theories it can very well be a probabilistic explanation.Thus, we can clearly see that Husserl’s views of empirical science anticipatedby some decades discussions that dominated the philosophical scene for asubstantial part of the last century in the hands of the logical empiricists,Popper and others. Moreover, an important point made by Husserl, which isparticularly relevant nowadays, is that in the process of knowledge the dataunder consideration do not remain completely unchanged but are modified.This brief comment by Husserl could serve as the basis for multiple discus-sions: (i) as a first step of a philosophical justification of current quantumtheory; (ii) as the basis for a critique of empiristic philosophies which still arebased on sense data as ‘‘unrevisable hard facts’’ of experience; and (iii) as thestarting point of an epistemology based on already structured states of affairsas building blocks, an epistemology that with regard to mathematicalknowledge Husserl developed in the Sixth Logical Investigation.

138 Ibid.139 Ibid.140 See footnote 19 above, as well as the present author’s paper ‘‘The Structure of the Prole-gomena’’.

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Appendix141

In the recent literature on Husserl’s views on mathematics142 the tendencyalready mentioned in footnote 40 to associate Husserl with different variantsof constructivism, which, due to the apparent affinities of Husserl’s later tra-scendental phenomenology with Kant’s trascendental idealism would lookvery plausible, has experienced a sort of revitalization. However, as alreadymentioned, the fact of the matter is that both in his 1929 Formale undtranszendentale Logik and other recently published courses contemporarywith his trascendental turn or later,143 there is no sign of any constructivism.Husserl’s views on mathematics remained the same after the trascendentalturn. Although this is difficult to swallow by traditional phenomenologists, itrepresents no problem for those like the present author, who coincide withCarnap in seeing the phenomenological reduction as a purely methodologicaldevice. Thus, Platonism survives the trascendental turn.144

In our dissertation of 1973, however, we discussed an extensive manuscriptof Husserl, which we read in the Husserl Archives in Cologne, in whichHusserl seriously considered constructivism. The manuscript, titled ‘‘TheParadoxes’’ and with the inscription A I 35, consists of two parts, namely, parta, dated 1912, and b, dated 1920. In the 1912 part of the manuscript Husserl isconcerned with different ways to solve Russell’s paradox—or better: Zermelo-Russell’s—and similar paradoxes. Husserl bases his discussion on twoimportant points, namely: (i) Not every meaning is fullfilable in a possibleintuition; e.g., a round quadrangle can be thought, but cannot be intuited,there is no sensible or categorial intuition of it; (ii) One has to distinguishbetween different levels of language, thus, modifying a little Husserl’sexample,145 a proposition (or name) of a proposition S is of a level immediatlyhigher than S. We have here the nucleus of a theory of types. As Husserl

141 This Appendix , based on the last chapter of my dissertation—see references—, discusses a stillunpublished manuscript of Husserl, Manuscript AI 35, which I read in the Husserl Archives inCologne in 1971 or 1972. I hereby thank the Husserl Archives in Cologne for having allowed me toread the manuscript while I was working on my dissertation. So far as I know, Claire O. Hill is theonly other Husserl scholar that has referred in print—in her paper mentioned in the refer-ences—to this valuable manuscript.142 See, e.g., Richard Tieszen’s Mathematical Intuition.143 The manuscripts on which is based Einleitung in die Logik und Erkenntnistheorie date from1906–1907, thus, from the years of the transcendental turn. See on this issue the editorial intro-duction to Husserl’s Die Idee der Phanomenologie. The manuscripts on which is based Logik undallgemeine Wissenschaftstheorie are of the transcendental phenomenology years144 Although, as pointed out in footnote 40, constructivism is perfectly compatible with tran-scendental phenomenology, one cannot say that it survived the transcendental turn—in the samesense in which one cannot say that a person born in 1960 ‘‘survived’’ the Second World War—,because it was never propounded by Husserl and certainly not before the transcendental turn.145 See A I 35, p. 11.

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makes it clear,146 the Russell set would simply be a countersense. MoreoverHusserl argues147 that membership in a set is an example of what he callsrelations of essences, and in such relations the members cannot be identical.Beginning on p. 13 of the manuscript, Husserl speaks also about sets con-structible from the axioms and definitions. As Husserl points out,148 oneshould not ascribe an extension (or set) to all general concepts. As an examplehe mentions149 that the mere something in general of formal ontology, onwhich all formal ontological fundamental concepts are based, does not haveany extension. Husserl considers other less palatable solutions in the first partof the manuscript, like the possibility of considering the notion of set as aspecial case of the notion of a whole, but what is important is that, as he stateson p. 25 of the manuscript,150 from the fact that we can speak about all setsdoes not follow that the totality of sets is a set, in the same sense that from thefact that we can speak about all possibilities does not follow that the totality ofall possibilities is a possibility.

Contrary to what I sustained in my dissertation, and even though Husserluses the expression ‘‘contructible’’ on pp. 12–13, the whole discussion ofHusserl in part a of the manuscript is perfectly compatible with his philosophyof mathematics as presented in Logische Untersuchungen, especially if weconsider his epistemology of mathematics of the second part of the SixthLogical Investigation, in which he offers an iterative constitution of mathe-matical objects in categorial intuition. Such a view is clearly related to theviews of his friends Cantor and Zermelo on the iterative notion of set, which isnot to be related with constructivisms of Kantian or Brouwerian, or any othersort.

The case of part b of the manuscript is somewhat different. It dates from1920, two years after the publication of Hermann Weyls Das Kontinuum.Weyl and his wife had been students of Husserl and were life-long friends ofhim. It seems that the publication of Weyl’s book, in which a mild construc-tivism was propounded, as well as the personal contact with his much youngerfriend, exerted a momentary influence on Husserl, which reflected itself in thesecond part of the manuscript. In this manuscript Husserl tries to show thatthe way to avoid the paradoxes consists in a constructive axiomatization of settheory. More explicitly, he stresses that a manifold is to be understood as a‘‘constructively (definite) characterized region of objects, which remains(materially) undetermined, whose objects can be constructed by the iterationinto infinity of definitely formed operations, and whose axioms must be sochosen as to found a priori such constructibility.’’151 Thus, for Husserl in part

146 A I 35, p. 12.147 Ibid.148 Ibid., pp. 17–18.149 Ibid., p. 17.150 A I 35, p. 25.151 Ibid., pp. 47–48.

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b of the manuscript, the doctrine of manifolds should be ‘‘the mathematicaldiscipline of the possible constructible infinities’’ and its task should be ‘‘toconstruct a priori the possible forms of such infinities as constructive sys-tems.’’152 With respect to Russell’s—better Zermelo-Russell’s—Paradox,Husserl says153 that it should not be assumed that concepts like that of the setof all sets that do not contain themselves as members have a totality, i.e., a setas extension, and that what such a paradox shows is that there is still no logicof sets in general. Moreover, Husserl adds154 that sets should be demonstrablyconstructible with respect to all its members, and that mathematics mustfurnish an existence proof of each and every set. Husserl is, however, notexplicit enough with respect to his notion of constructibility. It is only clearthat he requires an existence proof of each set. An interesting question here iswhether Husserl’s theory of manifolds and, in general, of mathematical ob-jects required some revision on the basis of these constructivistic leanings of1920, since, as we have shown elsewhere,155 neither Russell’s nor Cantor’s setscan be obtained in the iterative hierarchy of mathematical objects propoundedin the Sixth Logical Investigation. The fact of the matter is that in his laterFormale und transzendentale Logik there is no explicit mention of such arestriction to constructible manifolds. Thus, either Weyl’s impact on Husserlwas of short duration or he was convinced that his original philosophy ofmathematics, together with his epistemology of mathematics, in which hisiterative hierarchy of mathematical objects is inserted, were enough to pre-vent the paradoxes. These alternative explanations are by no means exclusive,and most probably both are correct.

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