𝟐.𝟐 ℑ𝔫𝔱𝔲𝔦𝔱𝔦𝔳𝔢𝔫𝔢𝔰𝔰 𝔞𝔫𝔡 𝔩𝔬𝔤𝔦𝔠𝔦𝔰𝔪

 𝔗𝔥𝔢 great British empiricist Hume, in this epistemological works, emphasized the fallibility of empirical knowledge. Perhaps for this reason, one of Kant's basic presuppositions is that anything that is absolutely certain must not be empirical, but based solely upon the mental constitution of the subject. Necessary truth, including mathematical truth, must be a priori: truley mathematical propositions are always a priori, not empirical, judgements, since they involve necessity, which cannot be gained from experience.¹ 

David Hume 1711 – 1776

𝔄𝔰 we have seen, Kant regarded the world as a representation in the subject, partly determined by the impressions recieved from the "things in themselves," partly by a priori characteristics of the subject's sensitivity and conceptual abilities. As is well known space and time were not to be seen as traits of the external world, but as a priori forms of our sensitivity or intuition [Anschauung], that determine our representation of the world with absolute necessity. Mathematics develops the consequences that can be extracted a priori form or pure intuition, it develops the theory of the possible constructions within our forms of intuition, space and time. Having its origin in the subject, it is now possible to understand the necessity that accompanies mathematical knowledge. 

𝔎𝔞𝔫𝔱’𝔰 explanation was satisfactory as far as geometry was concerned, but his discussion of arithmetic and algebra was unsystematic and vague. An attempt to correct this situation can be found in William Rowan Hamilton, the best example of a Kantian mathematician, which is not to be found in Germany, but in Great Britain. In 1837 he published a paper, that went almost unnoticed, in which he regarded "algebra" as "the science of pure time" [Hamilton 1837]. This was a time when terminology was in flux: many mathematicians saw arithmetic, algebra and analysis as a unity, but the name used for that discipline varied greatly; some called it "analysis," Hamilton preferred "algebra," and, as we will see, several German mathematicians employed the term "arithmetic". Within the Kantian frame, Hamilton's conception gives rise to a quite satisfactory symmetric schema, in which geometry is the science of spatial intuition, and algebra the science of temporal intuition. Fifty years later, Helmholtz [1887] defended a version of that thesis when he proposed that the origins of natural numbers of are related to the perception of time. Besides that speculative vision, Hamilton presented very interesting mathematical ideas, including the notion that what is essential in ℝ is the presence of a continuous ordering, a detailed and quiet rigorous treatment of arithmetic, and the brilliant idea of introducing the complex nunbers as pairs of real numbers. It seems that, this time, the philosophical grab was an obstacle for the diffusion of interesting thoughts. Nevertheless, his mathematical and philosophical ideas become well known later, thanks to the preface to his lectures on Quaternions [Hamilton 1853].² 

August Ferdinand Möbius 1790 – 1868

𝔄𝔯𝔬𝔲𝔫𝔡 this time, the Kantian conception was quite influential in Germany. It was common to defend that all mathematical knowledge bears the mark of its intuitiveness [Anschaulichkeit]. A good example is given by the correspondence between Apelt, a philosopher follower of Fries, and thus a neokantian, and the mathematician Möbius. Apelt commented upon Grassman's Ausdehnungslehre that it was remarkable, but based on a false philosophy of mathematics. 

An abstract theory of extension [Ausdehnungslehre] as he is looking for, can only be developed out of concepts. But the source of mathematical knowledge is not in concepts, but in intuition. 

Ernst Friedrich Apelt 1812 – 1859

𝔐𝔬𝔟𝔦𝔲𝔰 replied that he had not been able to read much in Grassman's book because of its abstract character, and granted that intuition is the "essential trait" of mathematics.³ 

Once again, this is a version of intellectualism, but quite different from the former band, Here, the symbols and formalism are taken simply to denote those figures, numbers, etc. that are constructed in intuition. Such a viewpoint, that avoids enthroning the symbolic plane, and prefers to emphasize contentual aspects, is actually closer to the conception that won the day. Against the formalists Gaus, Couchy, and their followers would have contended that mathematics does not study symbols, but numbers, functions and the like. Symbols and operations only have a sense insofar as they represent objects and processes that seem to stand on a different plane. This is one, though only one, of the reasons why the names of Hindenburg and Ohm aquired a ludicrous ring late in the century.⁴ 

Carl Hindenburg 1741 – 1808

𝔗𝔥𝔢 Kantian epistemological conception seems to have been widely influential in German scientific circles, even becoming a kind of basic 'common sense' the emergence of logicist viewpoints late in the century, which is quite difficult to understand for present-day historians, can actually be better understood when seen in this context. The development of mathematics in the nineteenth-century showed a clear tendency away from the intuitive, and toward the abstract. The most common example is non-Euclidian geometry, the emergence of a variety of consistent, alternative theories, which had the effect of denying the character of being basic or intuitive to any one of them. One of Gauss's comments on this issue, in a letter to Bessel of 1830, employs a Kantian language for correcting Kant's opinion: 

According to my most intimate conviction, the theory of space has a completely different position with regards to our a priori knowledge, as the pure theory of magnitudes, Our knowledge of the former lacks completely that absolute convictions of its necessity (and therefore of its absolute truth) which is characteristic of the latter, we must humbly acknowledge that, if number is only a product of our minds, space also has a reality outside our minds, and that we cannot a priori prescribe its laws completely.⁵ 

Besides the example of non-Euclidean geometry, which only after 1860 began to affect the mathematical community as a whole, many more could be given. It suffices to recall the theory of real functions, the many anomalous examples that began to proliferate after the mid-century. Or algebraic examples, such as the quaternions and other new kind of numbers, and the abstract notion of group finding its place as an extremely useful tool for diverse applications. Or the evolution of number theory in the modern direction of algebraic numbers. 

𝔑𝔬𝔴, suppose that we are thinking with in a Kantian overall frame, and we are not willing to abandon the fundamental thesis that mathematics has its origins in the human mind, not in experience or the outer things. How can we possibly make sense of those changes? We need to take into account one more aspect of Kant's philosophy. According to the Königsberg philosopher, the a priori material of the human understanding does not simply consist in space and time in forms of intuition, it also includes a whole set of concepts or categories, that we systematically apply in categorizing the phenomena of the world. It suffices to take look at the index of Kritik der reinen vernunft, to see that Kant calls the doctrine of the forms of institution "aesthetics," and the doctrine of the categories and concepts of the understanding "[transcendental logic]," Gauss's thoughts on geometry cause doubt on the real existence of an inborn form of spatial intuition, since the problem of the geometry of real space is coming to be seen, to some extant at least, as an empirical issue. The post-Kantian philosopher Herbart, who counted some mathematicians among his followers, had already abandoned Kant's postulate of the forms of intuition for purely philosophical motives, and criticized it sharply (see his 1824 Psychologie als Wissenschaft in [Herbart 1964, vol 5, 428-29]). 

𝔏𝔦𝔨𝔢𝔴𝔦𝔰𝔢, even Hamilton's supposedly intuitive foundation of algebra in pure time involve many abstract or conceptual elements that cannot possibly be related to any simple intuition. Among them are his consideration of "steps" in time, of ratios between such steps, and of pairs previous elements [Hamilton 1837,1853]. All this suggests that the basic thesis of the intuitiveness [Ansachaulichkeit] of mathematics ought to be abandoned. On the other hand, the 19th-century development from the intuitive to the abstract confirms that mathematics has much more to do with pure concepts than was previously thought. To abandon the reference to intuition, while sticking to the idea that mathematics is a priori, is to the consider mathematics as a theoretical development based solely upon concepts of the understanding. That is to say, in Kantian terminology, as a development of 'logic.'

ℑ should make it explicit that the previous exposition has simplified some important aspects, like the complex issue of the relations between "formal logic" (Kant's term) and "transcendental logic." But it is not my purpose to produce a philosophical analysis of Kantianism. Although Frege considered himself a Kantian, I am not trying to defend that anyone of the logicists was an orthodox Kantian. More interesting for present purposes is to considered the epistemological frame that was implicitly accepted by German scientists during the nineteenth-century. This seems to have been closely related to Kantainism, but not in an orthodox reading. Rather, scientists freely mixed some Kantain elements with several ideas taken from the sciences themselves. To give just an example, in the nineteenth-century it was quiet common to think about human reason, instead of trying to be faithful to Kant's pure reason - following Fries, It was common to adopt a psychologistic reading of Kantian philosophy. Actually, in most cases the Kantian elements that can be found in a scientist or mathematician may be explained as coming through other scientist's work, rather then from a careful reading of the philosopher. 

Gottlob Frege 1848 – 1925

𝔚𝔥𝔞𝔱 is important, then, is that several mathematicians understood the abstract turn, and quite specifically, as we will see, the set-theoretical reformulation, as implying that mathematics is a development of logic [Frege 1884; Dedekind 1888].⁶ 

𝔒𝔣 course, to become a logicist was not simply to apply some Kantian thesis. By the late nineteenth-century, a serious occupation with logicist ideas meant to give a clear formulation of formal logic that might be seen as sufficient for founding mathematics, and this implied the need to go beyond received logical theory. We will pay attention to this issue later, particularly in chapters VII and X. At this point, however it is important to emphasize that our present image of logicism is taken from the writtings of Russel and his followers, which makes us lose historical perspective, since the epistemological frame and quite specifically the notion of logic changed decisively in the period 1850 – 1940. Initially, logicism was typically a German trend that makes full sense against the background of a 19th-century epistemology permeated by Kantian presuppositions. Logicism was a reaction against the specific Kantian theory of the origins of mathematics, a reaction based upon, and favoring, the abstract tendencies that became quiet evident after the mid-century.⁷ 

Bertrand Russell 1872 – 1970

𝔄𝔰 will be seen throughout this book, beginning with chapter II, most mathematicians and logicians in the second half of the nineteenth-century took the notion of set to be simplify a logical notion. Actually the logicist programm would have been most implausible, for technical reasons, unless logic embraced some kind of set theory. During the early decades of our century, as a consequence of the set theoretical paradoxes, the panorama changed radically. The paradoxes meant a revolution in he conception of logic; part, if not all, of set theory 'divorced' from logic, and the logicist program suddenly lost its plausibility. Subsequent changes in logical theory, which led to the wide acceptance of first–order logic as the main example of a logical system, even deepened the gap that separates us from 19th-century authors. The concept of 'Logic' is also a historical one, and any attempt at understanding the emergence of logicism – directly related to the history of set theory – must be quite aware of the radical transformation which that concept underwent from 1850 to 1940.

1. [Kant 1787, 14]: "eigentliche mathematische Sätze jederziet Urtheile a priori und nicht empirisch sind, weil sie Nothwendigkeit bei sich führen, welche aus der Erfahrung nicht abgenommen werden kann."

2. both Cantor [1883, 191-92] and Dedekind [1888, 335] critisized this conception. 

3. Quoted in [Grassmann 1894, vol. 3, part 2, 101-2]. 

4. Mittag-Leffer wrote in 1886: "Kronecker emploie toutes les occasions à dire du mal de Weierstrass et de ses recherches. II desait même l'autre jour en parlant de lui et Weierstrass que Gauss était peu connu et peu estimé de ses contenporaines, tandis que Hindenburg était le grand géométry populaire de ce temps en Allemagne" [Dugac 1973, 162] On Ohm see [Bekemier 1987, 77-82]. 

5. [Gauss 1863/1929, vol 8, 201]: "Nach meiner innigsten Überzeugung hat die Raumlehre zu unserm Wissen a priori eine ganz andere Stellung, wie die reine Grössenlehre; es geht unserer Kenntniss von jener durchaos diejenige vollständige Überzeugung von ihrer Nothwendigkeit (also auch von ihrer absolute Wahrheit) ab, die der letztern eign ist; wir müssen in Demuth zugeben, dass, wenn die Zahl bloss unseres Geistes product ist, der raum auch ausser unserm Geiste eine Realität hat, der wir a priori ihre Gesetze nicht vollständing vorschreiben könen". 

6. The case of Riemann is different. He was a perfect example of the abstract trend, and some of his statements could incite logicist conclusions, but he was a careful philosopher and a follower of Herbart, Now Herbartianism avoids all apriorism, and therefore it is incompatible with logicism (see § II.1–2). 

7. This topic will be taken up in Chapter VII. 

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