Infinity (Stanford Encyclopedia of Philosophy)

Infinity is a big topic. Most people have some conception of things that have no bound, no boundary, no limit, no end. The rigorous study of infinity began in mathematics and philosophy, but the engagement with infinity traverses the history of cosmology, astronomy, physics, and theology. In the natural and social sciences, the infinite sometimes appears as a consequence of our theories themselves (Barrow 2006, Luminet and Lachièze-Rey 2005) or in the modelling of the relevant phenomena (Fletcher et al. 2019). Mathematics itself has appealed to some form of infinity from its beginning (infinitely many numbers, shapes, iterated addition or division of segments) and its contemporary practice requires infinitary foundations. Any field that employs mathematics at least flirts with infinity indirectly, and in many cases courts it directly.

Philosophy countenances infinity in myriad ways, either directly or indirectly, in most of its sub-fields—here is a tiny sample taken from the contemporary discussion (we shall discuss historical material in Section 1 and in Section 2, and many further examples in later sections). Some metaphysicians contend that there are infinitely many possibilities/possible worlds and canvas how big this infinity is (e.g. Lewis 1986). Philosophers of religion debate whether the divine is infinite, whether the divine creation is infinite, and whether the value of the afterlife is infinite. Epistemologists debate whether there can be an infinite regress of justification, and if so, whether it is problematic (Klein 2000, Peijnenburg 2007, Atkinson and Peijnenburg 2017). Formal epistemologists traffic largely in an infinitary notion of ‘probability’ (more in Section 6). Population ethics for infinite populations is a lively topic, and they are thought to pose distinctive problems for consequentialism (Nelson 1991). Social and political philosophy appeal to the notion of convention, often thought to involve ‘common knowledge’, with a putative infinite hierarchy of mutual knowledge (Lewis 1969). Philosophers of language and mind grapple with problems that infinitary operations such as ‘plus’ create for meaning and rule-following (Kripke 1982), and whether language itself, or minds themselves, can be infinite (Nefdt 2019). Philosophers of mathematics debate whether stipulations that imply the existence of infinitely many objects can be said to be analytic (Boolos 1997, Wright 1999) and whether criteria of identity for infinite numbers must necessarily be Cantorian (Mancosu 2016). See Section 4. Concerns about infinity (and human finitude) appear in continental philosophy, not only in its 19th century historical sources (e.g., among others, Fichte, Schelling, Hegel, Kierkegaard, and Nietzsche) but in contemporary developments as well (e.g., among others, Heidegger 1929, Levinas 1961, Adorno 1966, Foucault 1966, Deleuze 1969, Badiou 2019). This list can be continued, if not ad infinitum, then ad nauseam.

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