LLOYD P. GERSON | 45 What are the Objects of Dianoia? Lloyd P. Gerson Department of Philosophy University of Toronto [email protected] ABSTRACT In this paper, I examine the problem of the so- called Mathematical Objects within the context of the Divided Line. I argue that Plato believes that there are such objects but their distinctness and the mode of cognition relative to them can only be understood in relation to the superordi- nate, unhypothetical first principle of all, the Idea of the Good. The objects of mathematics or διάνοια are, unlike the objects of intellection or νόησις, cognized independently of the Good. https://doi.org/10.14195/2183-4105_18_4 46 | What are the Objects of Dianoia? Aristotle’s testimony that Plato introduced the objects are to be made according “mathematicals” (τὰ μαθηματικά) in between to the criteria of clarity (σαφηνεία) (μεταζύ) Forms (τὰ εἴδη) and sensibles (τὰ and truth (ἀλήθεια). The first of αἰσθητά) raises a host of problems, the vari­ these criteria suggests the possibil­ ous solutions to which are no doubt familiar ity of continuous gradation; on first to everyone here.1 I suppose that any effort to glance the second does not. Never­ move the discussion forward has to start with theless, if truth here is “ontological” some indisputable claims made by Plato es­ truth and not “semantic truth,” then pecially in the Divided Line passage. I shall gradation may well parallel that of begin simply by listing the claims that seem clarity. I understand ontological truth to me to be indisputable or nearly so and then as a relational property of Being, the move on quickly to highly disputed territory. property of being intelligible or trans­ Here are the claims that in my opinion should parent to our intellects. So, objects of shape our reflections on the putative status of cognition can be more or less clear mathematicals: and more or less true, that is, more or less intelligible to us. That means that 1. In the Divided Line, Plato makes a there is a gradation of intelligiblity hard distinction between the intel­ according to the criterion, roughly, ligible world and the sensible world, of transparency to an intellect alone. between that which is available only to The sensible world is less intelligible thought and that which, though avail­ than the invisible world because the able to thought, is so only on the basis former is enmeshed in that which is of sense ­perception.2 unintelligible or opaque to an intellect 2. There are a number of correlations or (as opposed to the senses). improper proportionalities in the Di­ vided Line: just as (A) the mode of cog­ According to the description of διάνοια nition for the top section (ἐπιστήμη) and its objects along with Aristotle’s testi­ is to (B) the mode of cognition for the mony, we might conclude that διάνοια has as bottom section (δόξα), so in the top its objects mathematicals, whereas νόησις has section, (A1) the mode of cognition as its objects the paradigms of these: instead in the first part of the top section of multiple triangles, Triangularity; instead of (νόησις) is to (A2) the mode of cog­ multiple numbers, Form­‑Numbers.4 When the nition in the second part (διάνοια), as mathematician says, “let there be a triangle,” or in the bottom section, (B1) the mode “take a prime number,” using sensible images of of cognition of the first part (πίστις) these, he is striving to achieve cognitive success is to (B2) the mode of cognition of the regarding Triangularity or Number. Since such second part (εἰκασία).3 success does not turn διάνοια into νόησις, we 3. The distinction between each of the need to ask first why this is so. objects of the four modes of cognition At A1, in order to have νόησις, one must is as irreducible as are the distinctions “ascend” to the Good and then “descend” among the four modes of cognition through Forms, presumably back to the Form themselves. The distinctions among we started with. Why, if the mathematician LLOYD P. GERSON | 47 provides a proof of a theorem in geometry, is necessary for knowing the Forms for one very this not enough? That is, why should the ascent good reason. If “good” is a property of a Form, and descent be so significant that it leads Plato we cannot know that the Form is good unless to distinguish two modes of cognition, one not and until we know what that Form is. This is of making the ascent and descent and one mak­ course the principle laid down in Euthyphro.6 ing it?5 Why is there more clarity and truth in So, if we have to “ascend” to the Good in order νόησις than in διάνοια? to know what a Form is, the claim being made is In order to be able to answer these ques­ not that we need to be able to see that the Form tions, let us make concrete the mathematical is good or, stated otherwise, that it is good that posits. Assume there is a Form of Three and the Form be what it is. It is not the predicative a Form of Four. Supposedly, the Idea of the nature of goodness that is at issue, so to speak, Good somehow makes these knowable. Since but the metaphysical principle, the Good. the Forms are not composed of units, they are The opacity of the claim that the Good is not addable. It is not the case that the Form of the source of the knowability of Forms and Three and the Form of Four when added are the that, in addition, the Forms cannot be known Form of Seven. Nevertheless, when the math­ without “ascending” to the Good evidently de­ ematician adds the numbers three and four, feats the ingenuity of those scholars who try to he gets seven. And it is something about the understand what is going on here while staying Form of Three and the Form of Four and the within the confines of Republic.7 By contrast, Form of Seven that guarantees that this is an the entire Platonic tradition up to the begin­ eternal and necessary truth. Stated otherwise, ning of the 19th century was in accord in taking what is needed is a metaphysical explanation as evident that we needed to have recourse to of mathematical necessity. This explanation Aristotle’s testimony about Plato’s philosophy would tell us why a mathematician when do­ and the testimony of the indirect tradition in ing mathematics does not invent mathematical order to make any progress.8 Aristotle and the truths but discovers them. indirect tradition of Platonism do not suppose So, the Form of Three and the Form of Four that it is the Good as such that explains the are to be eternally related such that three plus relatedness of Three and Four; rather, it is be­ four is seven is an eternal necessary truth. Ac­ cause the Good is the One that we can see how cording to the text, it is the Idea of the Good the first principle of all unifies the Forms and that enables us to see this relation, and appreci­ thus explains the eternal relatedness of Three ate that it is other than the relation of addition, and Four and all the rest of the Forms. given that Form­‑Numbers are not addable. It The One is certainly not the number one. is, I think, nonsense to maintain that it is the Nor is it a unit. The most perspicuous way of goodness of the Idea of the Good that enables us describing it is as a uniquely absolutely simple to see this relation, that once we see that there explanatory principle. That is, it is unquali­ is an Idea of the Good, we can then see that fiedly incomposite. There can be no more than Three and Four are eternally related such that one unqualifiedly incomposite “things”; in ad­ they guarantee that when we add three and four dition, there can be no less than one unquali­ we get seven. It may well be that in some sense fiedly incomposite “thing”. This is evident by it is good that this is so. But it is unintelligible inspection (though also derivable from the (to me anyway) that knowing that this is so is second hypothesis of the second part of Par‑ 48 | What are the Objects of Dianoia? menides). If there were, counterfactually, two to the way that white light is virtually all the absolutely simple first principles, they would colors of the spectrum or the way in which have to differ in at least one property. But if a function is virtually its domain and range. either of the relata has a property, it is not sim­ It is one thing to grasp a function in its uni­ ple. So, there cannot be more than one. There versality. It is quite another thing to grasp a must be at least one absolutely simple explana­ manifestation or application of that function tory principle because without this, we could as such. And it is yet another thing to grasp never arrive at an adequate or sufficient expla­ the manifestation without knowing that it is a nation for anything (the τι ἱκανόν) of Phaedo. manifestation of that function. This last cogni­ We would, ex hypothesi, be left with something tive achievement is what children do in basic complex, something whose very complexity is arithmetic before they learn algebra. It is also, left unexplained. The relevant complexity at by the way, what computers do when they ap­ issue for Plato is that of the existence of some­ ply the function as a rule without grasping the thing with a specific nature.