𝟐. 𝔗𝔯𝔞𝔡𝔦𝔱𝔦𝔬𝔫𝔞𝔩 𝔞𝔫𝔡 '𝔐𝔬𝔡𝔢𝔯𝔫' 𝔉𝔬𝔲𝔫𝔡𝔞𝔱𝔦𝔬𝔫𝔩 𝔙𝔦𝔢𝔴𝔭𝔬𝔦𝔫𝔱𝔰

𝔗𝔥𝔢𝔯𝔢 have always been many possible approaches to the philosophical problems raised by mathematics – empiricism, Platonic realism, intellectualism, intuitionism, formalism, and many other intermediate possibilities. From what we have seen, one many expect to find, among 19th-century German mathematicians, an influence of philosophical ideas and a greater speculative tendency then among their foreign colleagues. Interesting early examples that will not be discussed in detail here are those of Bolzano and Kummer. But more relevant for our purposes is the fact that the influence of philosophy seems to have led to an increase of intellectualist view points in 19th-century Germany. 

Immanuel Kant 1724 — 1804

𝔎𝔞𝔫𝔱𝔦𝔞𝔫 philosophy has always been more congenial to scientists then idealism, so it is not surprising that during and after the acme of idealism it retained an importation status among them. A characteristic Kantian idea is that the subject (the philosophical I) enjoys a central position in the world, or at least in our knowledge of the world. The world is regarded as a representation, a set of phenomena that unfold in the screen of consciousness. Such phenomena are not simply determined by the stimuli coming from external objects, but also by some intrinsic characteristics of the subject's mind. This situation of the representational world as codetermind by subject and objects may be seen as parallel to that of scientific theories, when viewed from the hypothetico-deductive standpoint, since theories are codetermind by mathematically stated hypotheses and laboratory experiences. If we accept this parallelism, and take the conclusion that arises from it, we should come to think that mathematics is on the side of the subject. It should be related to the intrinsic characteristics of the subject or, to use Kant's phrase, to the a priori in the subject's mind, and we thus arrived at an intellectualist's conception. 

ℑ𝔫𝔱𝔢𝔩𝔩𝔢𝔠𝔱𝔲𝔞𝔩𝔦𝔰𝔪 can be found in association with several different conceptions of the foundations of mathematics in early 19th-century Germany. Two examples are Kantian intuitionism, and also a brand of formalism linked to the so-called natorial school. Let us begin with the later. 

𝟐.𝟏. 𝔉𝔬𝔯𝔪𝔞𝔩𝔦𝔰𝔱 𝔞𝔭𝔭𝔯𝔬𝔞𝔠𝔥𝔢𝔰

𝔄𝔯𝔬𝔲𝔫𝔡 1800, the combinatorial tradition was very influential amoung German mathematicians.¹ This trend was headed by Leipzig professor of physics and mathematics Carl F. Hindenburg, from 1794 to 1800 edited the first periodical devoted to mathematics to Germany, Archiv ƒür die reine und angewandte Mathematik, a journal that he used for the promotion of his conception of mathematics. Combinatorialists saw themselves as heirs to Liebniz, who had written about the ars combinatoria as a "general science of formulas" providing general combinatorial laws, that would embrace algebra as a subdiscipline [Liebniz 1976, 54–56]. Hindenburg and his followers regarded combinatorial theory as the core of pure mathematics and the basis for the theory of series, which they saw in turn as the foundation of analysis. Thus, the centeral themes for this tradition were issues in pure mathematics, including the much debated problem of the foundations of the calculus. Analysis became, for them, a theory concerned with the transformations of finite and infinite series of symbols, transformations that could be analyzed combinatorially.² 

Gottfried Wilhelm Liebniz 1646 – 1716

𝔗𝔥𝔢 combinatorial approach was not far from contemporary viewpoints, such as Lagrang's formal conception of the calculus in his 1797 Théorie des ƒonctions analytiques, and the related development of calculus of operations [Koppelmann 1971]. One may say that combinatorialism developed some trends that were clearly present in 18th-century mathematics, trends which would also lead to the British tradition of symbolical algebra [Knobloch 1981; Pycior 1987]. An influential formulation of ideas related to those of the combinatorialists was given by the Berlin professor Martin Ohm, brother of the famous physicist and a figure of some importance for what folows. His work was particularly successful among Gymnasium teachers and also, one many conjecture, among self-taught mathematicians. Ohm clearly formulated the program of basing all of mathematics upon the notion of natural number, a program that can also be found in his colleague Dirichlet in Kronecker, Weierstrass and Dedekind. According to Ohm [1822, vol. I, xi – xiii] only natural numbers have a real existence. While the rest of mathematics can be seen as a theory of numerical signs. Ohm's reconstruction of pure mathematics was largely based upon the manipulation of formulas in accordance with the algebraic rules, but on the basis of a purely analogical justification – an inheritance of the combinatorial tradition, and a trait he has in common with British symbolic algebra. That viewpoint made it possible to establish the use of divergent series on a sound basis, thus rescuing a peculiar characteristic of 18th-century analysis that would be severely criticized by Ohm's great contemporaries Abel, Cauchy and Gauss.³ 

Martin Ohm 1792 – 1872

𝔗𝔥𝔢 combinatorial approach can be labeled a purely formalistic viewpoint, since it regarded mathematics as a symbolical or syntactic construction. Ohm's approach seems to have been only partly formalistic, since he accepted the natural numbers as given object with their characteristic properties. But one might think that even this partly formalistic standpoint ought to be radically opposite to the intellectualist tendencies. Symbols however, may be taken to have primarily a mental existence, and this move changes the picture completely. In defining a formal power series, Oham says that it is a function of identifinitly great degree, and, "therefore, an entire function that is never really representable, but only lives in the idea within our-selves" [Ohm 1855, 239]. Moreover, what is essential in the calculus, according to Ohm, is not numbers but operations, i.e., "actions of the understanding" — where understanding [Verstand] is a characteristically philosophical, and more specifically Kantian term: 

In the most diverse phenomenon of the calculus (of arithmetic, algebra, analysis, etc.) the auther sees, not properties of quantities, but properties of operations, that is to say, actions of the understanding... It turns out that one only calculates with "forms," that is, with symbolized operations, actions of the understanding that have been suggested... by the consedration of the abstract whole numbers.⁴ 

𝔗𝔥𝔢 general symbolic rules, therefore, represent mental actions performed on mentally existing forms or symbols. Ohm's mention of operarions that are suggested by the consideration of whole numbers is a reference to the so-called "principle of permanence of formal laws" which is also charecteristics of British  symbolical algebra, and can be found much later in the work of Hermann Hankel [1876]. 

ℭ𝔬𝔪𝔟𝔦𝔫𝔞𝔱𝔬𝔯𝔦𝔞𝔩𝔦𝔰𝔪 was very successful until about 1810, but text books of that orientation continued to be published up to the mid-century. Ohm's approach, on the other hand, seems to have been widely influential among Gymnasium teachers and those who were more or less self-educated in mathematics.⁵ As late as 1860, the Göttingen professor Mortiz A. Stern published a textbook in which formalistic conceptions akin to Ohm's were central (see[Jahnke 1991]), several ideas strongly reminiscent of Ohm's can be found in early writings of Dedekind, perhaps coming through his teacher Stern, and in Weierstrass, apparently trough his teacher C. Gudermann [Manning 1975, 329 – 40]. 

An intellectualist conception of mathematics, couched in the language of 'forms,' can also be found in Hermann Grassmann. Grassmann [1844, 33] began his work by elaborating a general theory of forms [Formenlehre] as a frame for mathematics. The transition from one formula to another he regarded as a strictly parallel to a conceptual process that should happen simulateniously [op.cit., 9]. A form, or "form of thought," was simply an object posited by thought as satisfying a certain definition, a "specialized being generated by thought," And "pure mathematics is the doctrine of forms" [op. cit., 24]. This new, abstract conception of mathematics is related with the influence of several philosophers, from Liebniz to Schleiemacher, on Grassmann.⁵ Although he accepted the existence of an spatial intuition, Grassmann's Ausdehnungslehre was not dependent upon intuition, since it constituted the abstract, purely mathematical foundation for geometry, which is empirical. Despite similarities with the combinatorialists and Ohm, Grassmann abandoned excessive reliance upon the symbolical, and more specifically he abandoned reasoning founded upon analogy, thus going in the modern direction. Incidentally, it is worth nothing that his approach promoted, like no other in eaely 19th-century Germany, a formal axiomatic structuring of mathematical theories. 

Friedrich Schleiermacher 1768 – 1834

1. Although it has been customary to refer to this trend as the combinatorial school, following 19th-century usage, I shall prefer the word 'tradition'. We reserve 'school' for those institutional arrangements in which small groups of mature mathematicians pursued more or less coherent research programs, joined by a certain style or 'philosophy' of research (in the sense of Hawkins), traning advance students with which they worked side-by-side [see Gaison 1981, Servos 1993] On the other hand 'tradition' seems art to convey the idea of influence and community of interests and 'philosophy', but on a looser institutional, geographical and/or temporal basis [see the Introduction ]. 

2. On this topic, see [Netto 1908], [Jahnke 1987; 1990]. 

3. The fact that Ohm's approach was made by rigorous by the differentiation between symbolical and numerical equalities, and the rules of interpretation of the symbolical calculus, has been emphasize by Jahnke [1987]. See also [Bekemeier 1987]. 

4. [Ohm 1853, vii]: "In den verschiedensten Erscheinungen des Kalkuls (der Arithmetik, Algebra, Analysis, u. u.) erblicket der Verf. nicht Eigenschaften der Grössen, sondern Eigenschaften der Operationen, d.h. Akten des Verstandes...."

5. See [Lewis 1977] and [Otte 1989], An interesting, short analysis of Grassmann's mathematical method can be found in [Nagel 1939, 215–19].

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