NICHOLAS BAIMA | 41 The Problem of Intermediates, an Introduction Nicholas Baima Department of Philosophy Harriet L. Wilkes Honors College Florida Atlantic University nbaima@fau.edu In March 2018, Sophia Stone and I hosted a conference on the possibility of intermediates in Plato's dialogues at Florida Atlantic University Harriet L. Wilkes Honors College. The six pa- pers that follow this introduction were originally presented at the conference and were revised for inclusion in this volume. In our work, we aimed to preserve the "presentation feel" of the papers and thus they might not have the same degree of notes found in regular journal articles. Although taking this approach does come with some schol- arly costs, we believe it is more important and fruitful to preserve the "dialectical aspect" of the papers - we are Platonists, after all. The problem of intermediates is an expan- sive philosophical and interpretative issue. A proper study of it requires not only under- standing the Platonic corpus but interpreta- tions of it by Plato's students and followers as well. The problem begins - to somewhat of a surprise - with Aristotle. In Metaphysics Α, Aristotle asserts that ...besides sensible things and Forms he [Plato] says there are the objects of math- ematics, which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form itself is in each case unique. (987b14 -17; see also B.997b12 -24 and M.1076b39 -1077a9) Despite Aristotle's testimony, no place in Plato's corpus do we find an explicit endorse- ment of intermediate objects. Scholars, thus, face a choice: they can either accept the testi- mony and mine the corpus for places where Plato might implicitly endorse such a thesis or they can argue that Aristotle was confused and that Plato doesn't think intermediate ob- jects exist. https://doi.org/10.14195/2183-4105_18_3 42 | The Problem of Intermediates, an Introduction For most of history, students of Plato have opted for the former approach. Perhaps the most common place scholars have identified the intermediate objects described by Aristotle is in the Divided Line of the Republic. As we know, the line represents a parallel progression in both ontology and epistemology. Socrates carves up the line into four segments, with each corresponding to a unique affection in the soul (pathēma en tē psuchē): noēsis (knowledge) for the highest condition, dianoia (thought) for the second, pistis (belief) for the third, and ei‐ kasia (imagination) for the lowest (6.511d; cf. 7.533d -534a). Some scholars have argued that based on a principle defended in Book 5, each of these affections in the soul must be set over a different object. For in Book 5, Socrates dis- tinguishes knowledge (epistēmē), belief (doxa), and ignorance (agnoia) by arguing that different cognitive dunameis (faculties/capacities/pow- ers) are set over different objects (477a -478c). If this principle is accurate and if "affections in the soul" are dunameis, then it must be the case that dianoia is set over a different object from noēsis. Now because pistis (belief) corresponds to sensible objects and noēsis (knowledge) cor- responds to Forms, dianoia must correspond to something in between sensible particulars and Forms. When we add to this that dianoia deals with the intelligible (noēton) and not the vis- ible (horaton), and is explained by reference to mathematics (6.510b -e), it begins to look plau- sible that dianoia is set over the mathematical intermediate objects described by Aristotle. Starting from this position, in, "What are the Objects of Dianoia?" Lloyd Gerson seeks to explain the distinctiveness of dianoia, both in terms of cognition and ontology. Gerson argues that it is only in relation to the Form of the Good that an explanation can be found. Following the tradition of Platonism, Gerson argues that we must understand the Form of the Good as the One - the only unqualifiedly incomposite. Although everything is related to the One, it relates to nothing since this would make the One complex and thus no longer sim- ple. The objects of dianoia are the relational aspects or expressions of Being cognized in- dependently of the Good. However, since the late 19th century scholars have begun to challenge the legitimacy of Aris- totle's testimony. Several details of the divided line raise problems for the "old interpretation"; I'll focus on three. First, the divided line is the perfect place for Socrates to discuss inter- mediate objects, yet he never does; this makes it doubtful that he actually had these objects in mind. Second, when relating dianoia to the methodology of mathematicians, Socrates says that although the mathematician uses visible images and makes claims about them, they pur- sue their inquiry for "the square itself" (6.510d). This makes it sound like dianoia and noēsis deal with the same object. Third, provided that Plato actually accepts a principle that different dunameis (faculties/capacities/powers) are set over different objects, it doesn't follow that di‐ anoia is set over a different object because it is not described as a dunamis but an "affection in the soul" (pathēma en tē psuchē). It is true that different "faculties" correspond to different "affections," but we cannot infer a difference in faculty from a difference in "affection." Pain and pleasure are different affects, but they can correspond to the same faculty and object. Indeed, in, "Unclarity and the Intermedi- ates in Plato's Discussions of Clarity in the Re‐ public," Nicholas Smith argues that reconciling Socrates' discussion of ontology and epistemol- ogy in Book 5 with the account of the divided line is more problematic than scholars have thought. A large source of these problems stems from Socrates appearing to alter his descrip- tion of the line in Book 7 (533e -534a) from his NICHOLAS BAIMA | 43 initial account in Book 6 (509e -511e). Rather than clarifying the details of the Line, positing intermediate objects in the third segment only obfuscates things. Such an interpretation can neither make sense of the philosophical and proportional entailments of the line nor can it account for what Socrates says in Book 5 about Forms. In Book 5, we learn that epistēmē is set over "what is" (477b -478d) and "what is" is a perfect exemplar of whatever it is. The problem with mathematical intermediates is that they too are supposed to be perfect exemplars; thus, they qualify as "what is" by the standards of Book 5. In other words, there doesn't seem to be enough room for intermediates based on what Plato says in Books 5 -7. If this negative interpretation is correct, two questions arise. First, if dianoia is always oriented towards Forms and not intermediates, what is it about the mathematical inf lection of dianoia that prevents a clear apprehension of Forms? Second, if there is no evidence of intermediates in Plato's corpus, what explains Aristotle's testimony? Andy German's, "From Intermediates through Eidetic Numbers: Plato on the Limits of Counting," makes progress on both fronts. German argues that the opacity of dianoia results from its inability to grasp the inter -relation of Forms. In developing his argument, German draws on Jacob Klein's analysis of eidetic numbers in the Sophist. For Klein, the unsatisfactory attempt in the Soph‐ ist to understand the inter -relation of formal kinds on analogy to numbers reveals exactly how dianoia cannot grasp the foundations of its own activity. German concludes his paper by explaining how Plato and Aristotle are speak- ing about the same thing, but not saying the same thing about it. Emily Katz explores Aristotle's testimony from the other side of the coin in, "The Mixed Mathematical Intermediates." Katz seeks to ex- plain why Aristotle thought it was necessary that Platonists accept intermediates and what this can tell us about Aristotle's own commit- ments. Many scholars find Aristotle's criticism of Platonism's ontology as merely polemical (see especially, Metaphysics B.2.997b12 -24 and M.2.1076b39 -1077a9). Aristotle reasons that Platonists must be committed to (1) arithmeti- cal and geometrical intermediates, and if this is so, then they must be committed to (2) interme- diate sensible things, and if this is so, they must be committed to (3) intermediates of all sen- sible objects. Many feel that Aristotle is piling absurdities on Platonists; Katz, however, disa- grees. Katz argues that Aristotle's criticism is sincere because we find evidence of him voicing similar concerns elsewhere. This casts doubt on any interpretation of Aristotle's philoso- phy of mathematics that makes him fall prey to the objections he raises against Platonists. Sophia Stone's paper, "Monas and Psuchē in the Phaedo," shifts focus to the Phaedo. Stone explores the various meanings of key mathematical concepts used by Socrates in the dialogues. She argues that a proper under- standing of these mathematical concepts not only makes the final proof more plausible, but also demonstrates how there is room in Plato's ontology for the intermediate objects Aristotle attributes to him. The key idea is that monas (unit) and psuchē (soul) share a dual role: they can both exist embodied in sensible particulars and apart from them. This dual role is due to their ontological status of being in between Forms and sensible particulars. In his, "Thumos and Doxa as Intermediates in the Republic," Olivier Renaut reminds us that Plato's discussion of intermediates isn't only ap- plicable to metaphysics - after all, Plato is very much a philosopher of intermediates. Across varying subjects, ranging from ethics to cos- mology - and everything in between - Plato 44 | The Problem of Intermediates, an Introduction seeks to bridge and explain the gap between two poles. Renaut aims to explain the relation- ship between the psychological intermediate, thumos (spirit), and the epistemic intermediate, doxa (belief), in the Republic. Renaut directs our attention to three connections: (1) the ob- jects of thumos are always doxai, (2) thumos gives power to doxa to overcome the appetite, and (3) thumos gives doxa relative stability. In terms of education, these intermediates are mediations that not only give meaning to the positive pole but also help direct us to it. This volume is not intended to be the final word on Plato's intermediates, much more work still needs to be done. I hope that this volume not only advances the study of this issue, but also redirects focus on it. I thank all the con- tributors to this volume and participants at the conference, the Plato Journal for the opportu- nity to present this work, and Sophia Stone for her help with editing the papers and organizing the conference.