The Development of Mathematics and the Birth of Phenomenology
The article examines Husserl’s view of mathematics as a continuation of Weierstrass’s project. While Husserl adopts the more modern axiomatic approach to mathematics as opposed to Weierstrass’s genetic approach, he continues to be Weierstrassian in his preoccupation for foundational questions. The latter part of the article examines in what way the outcome is Platonistic in Husserl’s own usage of the term. By Platonism Husserl means both a belief in immutable and unchanging ideal structures, as well as, a search for critical justification in reflection. In the latter sense of the term Husserl’s “Platonism” can be seen as an outcome of Husserl’s Weierstrassian ethos.
In the foreword to the Prolegomena to the Logical Investigations (1900) Husserl writes that the Logical Investigations (1900–1901) was born from his attempts to clarify the philosophical foundations of mathematics (Hua 18, 5).1 Arguably, the most important discovery of Husserl’s Logical Investigations (1900–1901) is the notion of categorial intuition. In this chapter I will examine how Husserl’s engagement in the problems about the foundations of mathematics led him to the discovery of categorial intuition. Roughly, the story goes as follows: At first, following his teacher Karl Weierstrass, Husserl held that mathematics should be erected on the concept of number. Accordingly Husserl’s first philosophical works focused on the concept of number and the elementary arithmetic operations. During the latter half of the nineteenth century the mainstream approach to mathematics evolved from the Weierstrassian genetic approach into an axiomatic approach, to use terms introduced by Hilbert in 1900 (Ewald 1996, 1092–1093). Following the mainstream mathematicians Husserl adopted the modern axiomatic view of mathematics, according to which mathematics is about abstract structures and has little to do with numbers or counting. The development of mathematics thus produced new, independent, purely formal domains to be studied. The new ideal domains helped Husserl to develop his anti-psyhologistic point of view as well as led him to the discovery of categorial intuition in the Logical Investigations.
In the end I will briefly address the significance of categorial intuition to Husserl’s philosophy. The notion takes Husserl’s approach far beyond Kant, aligning Husserl’s approach rather with Aristotle, as has been hinted at by Heidegger (Heidegger 1985, 2000; Taminiaux 1985) and emphasized by Richard Cobb-Stevens (1990, 2002, 2003, 2004), and Jaakko Hintikka in the present volume. I will discuss the nature of Husserl’s Aristotelianism. I will argue that while Husserl’s approach towards reality is certainly Aristotelian rather than Kantian or Cartesian, Aristotelian approach has difficulties in including Husserl’s abstract view of mathematics into it. Indeed, Husserl’s view of logic has a closer ancient counterpart in Euclid, and through Euclid, Plato. I will distinguish two senses in which Husserl’s approach can be said to be Platonist: the first concerns his view about unchanging, identical mathematical objects, and the second relates to the demand for justification of the axiomatic theories. However, claiming that Husserl is a Platonist is not to say that Husserl was a metaphysical realist in a naïve sense (see Tieszen’s paper in the present volume). Husserl’s transcendental idealist interpretation of Plato is entirely in accordance with the non-metaphysical spirit of phenomenology.
1 Weierstrass and Mathematics as Rigorous Science
While the mathematical technique called “calculus” was invented by Newton, Leibniz, and others in the seventeenth century, only in 1821 Cauchy published the first systematic approach to analysis.2 The logical structure of Euclidean geometry set the standard of rigor for Cauchy. The result was the first rigorous definitions of limit, convergence, continuity, and derivative. Many of Cauchy’s discoveries were simultaneously arrived at by Bernhard Bolzano (1781––1848). But while Cauchy’s Cours was extremely influential, Bolzano’s work was relatively unknown until the latter half of the nineteenth century.
However, Cauchy still used expressions like “approach indefinitely,” or “as little as one wishes” that left room for ambiguities and invoked geometrical intuition. Ultimately the needed rigor was established in the 1860s, when Karl Weierstrass presented several results that considerably clarified several fundamental notions of analysis. By means of ɛ-δ definitions he could define for example the notion of limit by using only the real numbers, addition and “smaller than” relation, instead of using vague natural language expressions (e.g., variable “approaching” something). Because of the overall rigor of his writings he is generally considered to be the founder of modern analysis.
In his lectures in the 1870s and 80s Weierstrass demanded stepwise demonstrations of the basic notions of analysis, beginning with the concept of number and operations on the numbers (Dugac 1973, 64–65, 73, 78). Later, his conception of rigor matured into a demand for clarity by means of a detailed mode of representation while attempting to manage a chapter of science as a whole. Weierstrass’s student Gösta Mittag-Leffler has characterized Weierstrass’s ethos as a striving for an all-encompassing theory of the domain in question: “In one of his treatises Weierstrass expresses the conviction that the results he has attained ‘will at least interest those mathematicians who find satisfaction, when it is possible to bring a chapter of science to a genuine conclusion.’” (Mittag-Leffler 1897, 79)3
With these plain words Weierstrass himself characterized his whole activity and articulated the goal which he strived for in all his works. The history of mathematics will also support this, that until now no mathematician was able to reach this goal to a higher degree and in greater extent than he, and the goal is to bring complete chapters of the science into actual conclusion. (Mittag-Leffler 1897, 79)
Initially Weierstrass’s aim was to logically analyze the fundamental notions of analysis. While he wanted to articulate everything presupposed in the theory he also wanted to describe a branch of mathematics in its entirety. Thus Weierstrass’s ethos is characterized by an aspiration to descriptive completeness, to capture everything there is to say about the domain of a theory. This ethos may well have had an enormous influence not only on Husserl and his view of Definitheit (See below Section III, also Hartimo 2007), but also many others in his generation of mathematicians, in particular, on Hilbert’s Completeness Axiom.
2 Husserl in Weierstrass’s Footsteps
The great Weierstrass was the one who raised in me an interest for a radical grounding of mathematics during my student years in his lectures on the theory of functions. I learned to appreciate his efforts to transform analysis, which was so much a mixture of rational thought and irrational instinct and tact into a purely rational theory. He was after the original roots, attempting to postulate the elementary concepts and axioms to form a basis from which the entire system of analysis could be constructed and deduced with a completely rigorous, thoroughly insightful method. (Schuhmann 1977, 7)
Moreover, Malvine Husserl has reported that Husserl repeatedly said that he owes his scientific ethos to Weierstrass (Schuhmann 1977, 7; See also Becker 1970, 40).
To Husserl Weierstrass’s importance is in his demand for rigorous foundations for mathematics. Despite of his changing view of mathematics from the more genetic approach to the more axiomatic approach, in the sense of demanding foundations for mathematics, Husserl remained Weierstrassian for the rest of his life.
Today there is a general belief that a rigorous and thoroughgoing development of higher analysis …, excluding all auxiliary concepts borrowed from geometry, would have to emanate from elementary arithmetic alone, in which analysis is grounded. But this elementary arithmetic has, as a matter of fact, its sole foundation in the concept of number; or, more precisely put, it has it in that never-ending series of concepts which mathematicians call “positive whole numbers.” … Therefore, it is with the analysis of the concept of number that any philosophy of mathematics must begin. (Husserl 2003, 310–311).
Husserl explicitly takes on the task of continuing Weierstrass’s program and providing foundations to Weierstrass’s approach. In Husserl’s Habilitationsschrift this meant providing an analysis for the concept of number. A similar view is also expressed in the introduction to Husserl’s Philosophie der Arithmetik, Psychologische und logische Untersuchungen published in 1891. There Husserl states, that “[p]erhaps I have succeeded in preparing the way, at least on some basic points, for the true philosophy of the calculus, that desideratum of centuries” (Hua 12, 7, Husserl 2003, 7). The aim is thus to provide foundations to calculus by an analysis of the concept of number. Moreover, the analysis should render the number thoroughly intuitive.
3 Philosophy of Arithmetic as an Analysis of the Concept of Number
Husserl’s first published work Philosophy of Arithmetic (PA) was originally supposed to appear in two volumes, but only the first one came out. The published work consists of two parts with the first four chapters of part one repeating the contents of Husserl’s Habilitationsschrift “On the Concept of Number,” according to Husserl, almost word for word (Hua 12, 8). The first part of the PA gives a psychological analysis of our everyday concept of number. Indeed, Husserl later claimed that also Weierstrass started his analyses from our everyday concept of number. Around 1930, in an unpublished manuscript, Husserl wrote that “Weierstrass admittedly started with concepts that are already given in the natural thinking life and the tradition. But they will not be accepted without hesitation, but only after deliberate proof, namely as intuitive in their meaning as clear and identifiable, like the individual 1, the operative construction of 1 + 1 etc. (addition), equality of the individuals, etc.”(B II 23 8 a–b, my emphasis).4
A psychological analysis is to Husserl an analysis of an experience of the presentation of a number and in particular an elucidation of its “origin.” Since our intellect and time are limited, we can have an intuitive understanding of only a small part of mathematics. In order to overcome the limitations of our intellect we make use of symbols to assist our thinking. This already takes place in such simple tasks as counting objects. Indeed, we know almost all of arithmetic only indirectly through the mediation of symbols. Accordingly the second part discusses the symbolic representation of numbers. As the subtitle Psychologische und logische Untersuchungen suggests, it contains logical investigations of the concept of number.
I had already acquired the definite direction with regard to the formal and a first understanding of its sense by my Philosophy of Arithmetic (1891), which, in spite of its immaturity as a first book, presented an initial attempt to go back to the spontaneous activities of collecting and counting, in which collections (“sums,” “sets”) and cardinal numbers are given in the manner characteristic of something that is being generated originaliter, and thereby to gain clarity respecting the proper, the authentic, sense of the concepts fundamental to the theory of sets and the theory of cardinal numbers. It was therefore, in my later terminology, a phenomenologico-constitutional investigation; and at the same time it was the first investigation that sought to make “categorial objectivities” of the first level and of higher levels (sets and cardinal numbers of a higher ordinal level) understandable on the basis of the “constituting” intentional activities, as whose productions they make their appearance originaliter, accordingly with full originality of their sense. (FTL, §27a)
The second part offers an independent view of arithmetic, basing it on the use of signs. In the first part Husserl had explained that we are capable to have an authentic intuition only of numbers up to about 12. However, if we count the objects by enumerating them, we are already relying on symbolic methods (Hua 12, 104–105, Husserl 2003, 109–110). Thus it is clear that arithmetic in general cannot be authentically given in intuition (Hua 12, 192). The question that Husserl wants to answer in the second part of the Philosophy of Arithmetic is how the rest of arithmetic is given to us.
Husserl’s answer to the problem lies in the complete parallelism between the system of concepts and the system of signs. The idea is to start from certain concepts, translate them into signs, and then to operate on the signs in accordance to given rules. The resulting sign is in the end interpreted as a concept. Thus Husserl’s conception of the arithmetica universalis is based on the notion of computation, and the belief in, in modern terms, completeness and soundness of the two parallel systems. In the end, the basis for arithmetica universalis is in the sense-perceptible signs. Husserl also presupposes the existence of purely formal concepts that correspond to the results of the computations, thus maintaining a view that is rather independent of the first part of the Philosophie der Arithmetik.
logical investigation of the arithmetical algorithm … and the justification of utilizing in calculations the quasi-numbers originating out of the inverse operations: the negative, imaginary, fractional and irrational numbers. Critical reflections on the algorithm repeatedly occasion a closer examination of the question whether it is the domain of cardinal numbers, or some other conceptual domain, that general arithmetic in the primary and original sense governs. To this fundamental question the Second Part of Volume Two is then devoted. (Hua 12, 7; Husserl 2003, 7)
In addition he also planned to develop a new philosophical theory of Euclidean geometry, possibly on the basis of Gauss’s work Anzeige der Theoria residuorum biquadraticorum, Commentatio secunda. However, the second volume never appeared. Ironically, Husserl later claimed that the analyses of the psychological part of the PA already represent phenomenologico-constitutional investigation, while the second part on the logical investigations caused him problems and underwent several changes (Husserl 1994, 490–491, Hartimo 2007).
4 Logical Investigations and the Axiomatic Approach
For the sake of the present argument I will not discuss Husserl’s development between the Philosophy of Arithmetic and the Logical Investigations in detail here. But roughly what happened was that Husserl’s investigations took him to the developing views of Mannigfaltigkeitslehre. He searched for a general framework in which one could examine individual theories and their relationships with each other. Wanting to find an approach to analysis free from geometrical intuition he was mostly interested in Hermann Grassmann’s Ausdehnungslehre, coordinate-free geometrical calculus. However later he claimed interest also to Riemann’s, Hamilton’s, Lie’s, and Cantor’s approaches. He worked in Halle as Cantor’s colleague. The two discussed each other’s work, and for example, Schröder’s attempt to deal with concepts of set theory by means of the algebra of logic (Schuhmann 1977, 52; Peckhaus 2004, 593–594).5
In his investigations Husserl followed the general trend in the mainstream mathematics. In the end of the nineteenth century mathematics developed from the Weierstrassian approach towards more abstract theories. By the 1880s Weierstrass, and with him, Berlin, lost the leading role in the world of mathematics. Göttingen with Klein and Hilbert in the lead assumed the role in the late 1890s. Around the turn of the century there were several competing “logics,” i.e., different kinds of theories of manifolds, which were different kinds of suggestions for the general framework in which we could examine mathematical theories. The search for rigor in the foundations of mathematics reached its point of culmination in Hilbert’s work. Incidentally, in 1901 Husserl moved from Halle to Hilbert’s Göttingen.
Hilbert’s work was a culmination in a trend to give analysis purely qualitative basis. To Hilbert, the ultimate rigor to analysis, as well as to the rest of mathematics, is given by axiomatization, not through arithmetization. Hilbert contrasted his axiomatic method to the genetic method, in which the number domain is generated from a number one. Instead, in the axiomatic method, the existence of the elements of the domain is presupposed, while the relationships between the elements are determined by means of the axioms. The axiomatic method then requires proving the completeness and coherence of the system. In 1900 Hilbert added the Completeness axiom to his axiomatization of the Euclidean geometry. With it he posited its categoricity, i.e., roughly that all of its interpretations are isomorphic to each other. A complete axiomatization has formally only one interpretation, and thus one could as well talk about tables, chairs, and beer mugs as points, lines, and planes. In such an axiomatization the elements of the theory are considered from a purely formal point of view and their material realization is entirely irrelevant.
Soon after having moved to Göttingen in 1901, Husserl gave a so-called Doppelvortrag in Göttinger Mathematische Gesellschaft. In it Husserl explained his view of a definite axiomatic system that justifies the usage of the imaginary numbers. Definiteness to Husserl captures the formal domain in the manner of Hilbert’s complete axiomatic system. I have discussed Husserl’s view of definiteness elsewhere hence I will here only summarize the main point of Husserl’s lectures. For Husserl a definite system is, like Hilbert’s complete axiomatization, a categorical theory that has only one purely formal model (Hartimo 2007). The definite axiom system is Husserl’s ideal of a pure logic. It is the basis for Husserl’s arguments against psychologism as it shows an undeniable existence of an objective and purely formal theory. To Husserl, the problem with the psychological logicians is that they cannot account for something like that and hence to Husserl their views lead to relativism and skepticism.
Perhaps even more importantly, to Husserl the idea of pure logic is a source of a new philosophical problem: how is this newly found formal domain given to us? “So erwächst die große Aufgabe, die logischen Ideen, die Begriffe und Gesetze, zu erkenntnistheoretischer Klarheit und Deutlichkeit zu bringen. Und hier setzt die phänomenologische Analyse ein” (Hua19/1, 9). Instead of the givenness of numbers and sets, Husserl’s problem is now the givenness of the theoretical structures. The idea of pure logic is purely formal, thus there can be no sensuous intuition of it. The task is now to give a descriptive analysis of the constitution of the ideality of the abstract domains rather than that of the givenness of different sizes of collections of objects.
It is worth emphasis that Husserl’s problem is the givenness of the theoretical structures; Husserl does not postulate them, but he sees his task to be to describe what is handed to him by the mathematicians. His attitude towards the existence of mathematical objects and structures is largely neutral: he does not question them nor does he postulate them. He simply describes what the sciences, in the present case mathematics, have found there to be. In terms of ontological questions Husserl’s approach is thus rather naturalist. But, unlike a typical naturalist, in order to understand the givenness of what there is, Husserl turns to a transcendental description of how the given is constituted.
5 Categorial Intuition
Categorial intuition is Husserl’s initial answer to the givenness of the pure logic. Intuition of something according to Husserl means direct givenness of that something. Sensuous intuition means givenness of simple objects. Categorial intuition, on the contrary, means givenness of categorial formations, such as states of affairs, logical connectives, and essences. When we see that paper is white, we do not only see paper and whiteness but also that the paper is white. For Husserl this shows that what is directly given to us in our experiences is not restricted to sense data. The world appears to us as meaningful and structured. In addition to simple objects we intuit certain “surplus,” which makes the world garbed with meaning.
As Heidegger points out, in formulating the notion of categorial intuition Husserl moves far beyond Kant. Heidegger only hints at Husserl’s Aristotelianism, but some later phenomenologists have emphasized Husserl’s close resemblance to Aristotle. (See in particular Cobb-Stevens 1990, 2002, 2003, 2004.) After the publication of the Logical Investigations, Husserl’s interest in categorial intuition grows into all-encompassing analyses of the constitution of the given. In 1907, Husserl introduced the term Wesensschau for what in the Logische Untersuchungen he called idealizing abstraction. Wesensschau is a special case of categorial intuition, and it refers to our capability to “see” identical essences. Jaakko Hintikka bases his view of Husserl as an Aristotelian philosopher especially on Husserl’s notion of Wesensschau.
6 Aristotle or Plato (and Which Plato)?
In a very general sense Husserl’s attitude towards the reality indeed is Aristotelian rather than Cartesian or Kantian. We intuit categories, which would be an oxymoron for Kant. Moreover, what we intuit is out there in the world. Intuition is not introspection.
But if we take into account the role of abstract structural mathematics to Husserl, some reservations are in order. The problem is that the role of Aristotle’s syllogistic as well as his writings in mathematics, was to offer an organon for empirical sciences. Aristotle did not thematize axiomatizations of geometry or arithmetics in themselves. Consequently Aristotle’s syllogistics is not abstract enough to be able to include Husserl’s view of abstract structures. Aristotle’s, or at least his students’, approach to logic was guided by a practical interest, and it was not founded by pure logic. The main point of Husserl’s Prolegomena is that logic as a practical discipline has to be justified by logic as a theory.
The Euclidean axiomatization rather than Aristotelian syllogistics has set the theoretical ideal for mathematicians ever since the ancient times. In this respect, Husserl follows the mathematicians: the Euclidean ideal sets the standard of pure logic to him, too. To Husserl Euclid systematized the axiomatic ideal presented in Plato’s Republic (Hua 7, 34–35). Indeed, in 1918 in a letter to Julius Stenzel Husserl called himself a “phenomenological Platonist” (Briefwechsel 6, 427–428), and in the 1920s he repeatedly mentioned Plato as the most important philosopher in the history of philosophy. He also claims that instead of Aristotle, Plato is the one who establishes the ideal of rationality and logic (Hua 7, 34).
7 Platonism of the Eternal, Self-Identical, Unchanging Objectivities
When Husserl’s Platonism is discussed in the secondary literature, it usually refers to a view about abstract objects or mathematical objects. Indeed Husserl writes: “In fact, one cannot describe the given phenomena like the natural number series or the species of the tone series if one regards them as objectivities in any other words than with which Plato described his ideas: as eternal, self-identical, untemporal, unspatial, unchanging, immutable” (Hua 30, 34). Husserl’s view about the abstract objects derives from Lotze and especially Lotze’s reading of Plato’s ideas (Hua 20/I, 297, Briefwechsel 6, 460, for a detailed discussion of Husserl’s indebtedness to Lotze, see Hauser 2003). To Lotze Plato’s ideas are self-same and eternal concepts that are objective. They are valid (within a web of logical theory) but they do not have the reality of Sein (Lotze 1928). Accordingly Husserl explains his plan to have been to take Lotze’s view of the ideal domain and place all the mathematical and a good part of the traditional logic into it. Husserl’s way of avoiding Lotze’s occasional psychologism was to use the “new mathematical logic” of the late eighteenth century. In particular, in a categorical theory mathematical objects have an objective existence independently of our activities of judging. They exist unter dem Blick der Theorie, as Jocelyn Benoist (2003) has put it. This is a kind of Platonism, but Platonism without hypostatising, ohne topos ouranios (Hua 19/1, 106), and hence not naïve Platonism discussed by Tieszen nor the kind of Platonism Dieter Lohmar objects to in his contribution, but a kind that is consistent with Husserl’s transcendental idealism. Moreover, Husserl’s Platonism is metaphysically neutral in the sense that it only describes the way in which mathematicians relate to their subject matter. A definite, i.e., a categorical theory defines abstract objects uniquely, and the philosophers’ task is to describe the givenness of these objects. Thus Husserl does not postulate the realm of abstract entities. Rather he describes what mathematicians show us there to be. And they show us structures.
8 Platonism as an Aspiration for Reflected Foundations
But there is more to Husserl’s “Platonism” than the above described view of the objectivity of the concepts. Indeed, Husserl refers to Plato in a much deeper sense when he opens his Formal and Transcendental Logic (1929) saying that “Science in a new sense arises in the first instance from Plato’s establishing of logic.” Husserl goes on to explain that we inherit from Plato the attempt to gain genuine knowledge that is fully justified by reflection. This sense of Platonism however is also a part of Husserl’s Weierstrassian inheritance and has motivated Husserl’s work from the beginning as the search for intuitively evident foundations for mathematics. It is only later that Husserl found the justification for exact sciences to be thematized also in Plato’s dialogues, in particular, the Republic. This sense of Platonism shows also as a striving to a complete theory about a subject matter in question. Early Husserl claimed that he owed his scientific ethos to his teacher Karl Weierstrass (Schuhmann 1977, 7). Above we mentioned how Weierstrass’s close student Gösta Mittag-Leffler has characterized Weierstrass’s ethos as a striving for an all-encompassing theory of the domain in question (Mittag-Leffler 1897, 79). In Husserl’s writings Weierstrassian/Platonist ethos turns into aspiration for theoretical completion, but the completeness in question is not that of any kind of a fixed system, but rather that of a research project. Because of it the investigation of evidences has to be included into the domain of logic. Thus Husserl’s quest for completeness takes him to a never-ending project of estimating the telos of logic and the transcendental examination of the way various layers of logic are given to us. The description of various evidences is to provide logic with Socratic self-examination. It is an investigation of sense, the outcome of which is increased understanding, and to quote Husserl who quotes Socrates in Plato’s dialogues, “a life truly worth living, a life of ‘happiness,’ contentment, well-being, or the like…” (FTL, 5). Thus Husserl’s call for a “foundation” or a “justification” for sciences should not be understood only in an epistemological sense but also in the sense of Socratic-Platonic examination. Thus the view of philosophy as a justifying reflection of the sciences ultimately relates to what Husserl sees the task of phenomenology to be in general. This is the sense in which Bernet, Kern, and Marbach relate Husserl’s idea of philosophy to the Socratic-Platonic examination of the absolute knowledge in relationship to the self-knowledge (1989, 4). This aspiration to “know thyself” is, in Husserl’s view, an infinite historical task.
Plato emphasizes time and again the role of mathematics. To him the way into philosophy goes through mathematics. In this paper I have discussed how a similar road took Husserl from Weierstrass’s lectures via the abstract structural mathematics in the end of the nineteenth century, to the knowledge theoretical problem of the givenness and justification of the abstract structures. All along Husserl remains Weierstrassian in his aspiration to ground mathematics by means of insight. However, the developing mathematics uncovered the purely formal structures, which gives Husserl the pure ideality. Husserl’s phenomenology is an answer to the problem of combining these two tenets, and in both of these respects Husserl found echoes in Plato.
Husserl opens the Logical Investigations with the following claim about its origin: “Die logischen Untersuchungen, deren Veröffentlichung ich mit diesen Prolegomena beginne, sind aus unabweisbaren Problemen erwachsen, die den Fortgang meiner langjährigen Bemühungen um eine philosophische Klärung der reinen Mathematik immer wieder gehemmt und schließlich unterbrochen haben. Neben den Fragen nach dem Ursprung der mathematischen Grundbegriffe und Grundeinsichten betrafen jene Bemühungen zumal auch die schwierigen Fragen der mathematischen Theorie und Methode. …” (Hua 18, 5). Husserl goes on to explain how he had realized that “das Quantitative gar nicht zum allgemeinsten Wesen des Mathematischen oder ‘Formalen’ und der in ihm gründenden kalkulatorischen Methode gehöre. Als ich dann in der ‘mathematisierenden Logik’ eine in der Tat quantitätslose Mathematik kennenlernte, und zwar als eine unanfechtbare Disziplin von mathematischer Form und Methode, welche teils die alten Syllogismen, teils neue, der Überlieferung fremd gebliebene Schlußformen behandelte, gestalteten sich mir die wichtigen Probleme nach dem allgemeinen Wesen des Mathematischen überhaupt, nach den natürlichen Zusammenhängen oder etwaigen Grenzen zwischen den Systemen der quantitativen und nichtquantitativen Mathematik, und speziell z. B. nach dem Verhältnis zwischen dem Formalen der Arithmetik und dem Formalen der Logik. Naturgemäß mußte ich von hier aus weiter fortschreiten zu den fundamentaleren Fragen nach dem Wesen der Erkenntnisform im Unterschiede von der Erkenntnismaterie und nach dem Sinn des Unterschiedes zwischen formalen (reinen) und materialen Bestimmungen, Wahrheiten, Gesetzen” (Ibid., 6).
I have discussed Husserl’s relationship to Weierstrass as well as his Philosophy of Arithmetic (1891) in more detail in Hartimo 2006.
In einer seiner Abhandlungen spricht Weierstraß die Überzeugung aus, dass die von ihm erhaltenen Resultate “wenigstens diejenigen Mathematiker interessieren werden, welchen es Befriedigung gewährt, wenn es gelingt irgend ein Kapitel der Wissenschaft zu einem wirklichen Abschlusse zu bringen.”
Weierstraß beginnt zwar mit Begriffen, die von dem natürlichen Denkleben und der Tradition vorgegeben sind. Aber sie werden nicht unbesehen hingenommen, sondern nach bewusster Prüfung aufgenommen, nämlich als einsichtige ihrem Sinn nach klare und identifizierbare, wie das einzelne 1 die operative Bildung des 1 + 1 usw. (Addition), Gleichheit von Einzelnen etc. (B II 23 8a-b)
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