This paper presents the view of Richard Routley (later Sylvan) that mathematical objects are non-existent. [Cf. \textit{R. Routley}, Exploring Meinong's jungle -- and beyond. Canberra: Philosophy RSSS, Australian National University (1980), Chapter 10 of which is reprinted as: \textit{R. Sylvan} ``The importance of nonexistent objects and of intensionality in mathematics'', Philos. Math. (3) 11, 20--52 (2003; Zbl 1048.03008).] It proposes a response to the so-called characterization problem that afflicts any sort of Meinongianism, that if any condition \(\varphi(x)\) defines an object \(a_\varphi\) such that \(\varphi(a_\varphi)\) is true, then all sorts of untoward results seem to obtain; the response is to allow these results but locate them in other, non-actual worlds, which might be incomplete or inconsistent or both. (Allowing such worlds is familiar from relevant and paraconsistent logic.) The author then defends Routley's view against a number of objections, including that his Meinongianism is just Platonism in disguise. If anything, he argues, Platonism is better construed as a sort of Meinongianism. (This paper serves to introduce the reprint of Routley's chapter mentioned above).