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Hegel's notion of 'bad infinity', as it appears in connection with his critique of Newton's infinitesimal calculus in Science of Logic, has not been much appreciated nor incorporated into the tradition of logic or the philosophy of mathematics. In fact, it shares the fate of most of Hegel's logical distinctions, which are considered obscure, unscientific and best left to Hegelians.1However, and despite their generosity toward the idiosyncrasies of Hegel's style, Hegelians too generally doubt his competence in mathematical matters, bracketing his remarks on the 'bad' nature of infinitesimals with those on 'negative electricity' or phrenology. 2
Only recently, following Pinkard's programmatic paper (Pinkard 1981), has a re-assessment of Hegel's philosophy of mathematics been established as a philosophically fruitful exercise for the philosophy of mathematics and, just as importantly, for the logical analysis of language. That such a re-assessment is taking place within the analytical movement might be interpreted as a by-product of analytical philosophy's ongoing recollection of its idealist roots. A further contributing factor is the increasing recognition that Hegel's knowledge of contemporary mathematics was anything but marginal (see Wolf 1986: 200 and Lacroix 2000: 298). Finally, as regards logical analysis, recent interpretations have shown that Hegel is not at variance with the methods of modern formal logic; rather, he is more radical than them. 3In light of these considerations, the revision of the concept of bad infinity is of particular importance both for the study of Hegel's thought and for the philosophy of mathematics. The main thesis of this paper is that 'bad infinity' is a pivotal logical concept, introduced in a specific mathematical context for the sake of a more radical logical analysis of knowledge and of its evolutionary as well as social nature. Because of this generality, 'bad infinity' can be applied to all fields of knowledge, including mathematical development after Hegel in which the concept of infinity played a significant role.
To show this I start by challenging a widespread view, expressed by authors from Bolzano to Pinkard, that Hegel's notions of bad and true infinity replicate the traditional distinction between potential and actual infinity, and that Hegel favours the latter, though in some mysterious,...