O Problema da Hipótese na Linha Dividida da República de Platão
The Problem of the Hypothesis in the Divided Line of Plato's Republic
Weriquison Simer Corbani
weriquison.corbani@ifes.edu.br
IFES – Instituto Federal do Espírito Santo
Recebido: 02/02/2025
Received: 02/02/2025
Aprovado:17/02/2025
Approved: 17/02/2025
Publicado: 06/03/2025
Published: 06/03/2025
Resumo
Ao lançar a imagem da Linha, no Livro VI da República, Sócrates diz que a alma passa por quatro estágios de investigação: suposição (εἰκασία), crença (πίστις), pensamento discursivo (διάνοια) e inteligência (νόησις). No terceiro estágio, embora a investigação já trate da hipótese das Formas, o tipo de operação intelectual que encontramos aí não permite avançar para o próximo, correspondente à Forma do bem. Isso ocorre porque pela via do pensamento discursivo, que envolve processos lógicos, não é possível alcançar o “princípio de tudo”. Há, então, que abandonar a investigação por hipóteses e fazer uso exclusivo de outro tipo de operação intelectual, não mais instrumental, mas intuitiva. É por essa razão que apenas pela via da nóesis, da visão direta, se pode chegar ao conhecimento do bem. Esse artigo faz uma análise da distinção desses dois tipos de operações de raciocínios presentes na Linha dividida.
Palavras-chave: hipótese, linha dividida, bem, Platão.
Abstract
When presenting the image of the Line in Book VI of the Republic, Socrates explains that the soul progresses through four stages of investigation: supposition (εἰκασία), belief (πίστις), discursive thinking (διάνοια), and intelligence (νόησις). In the third stage, although the investigation already addresses the hypothesis of the Forms, the type of intellectual operation found at this level does not allow advancement to the next stage, which corresponds to the Form of the Good. This is because, through discursive thinking, which involves logical processes, it is impossible to reach the “principle of everything.” It is therefore necessary to abandon hypothesis-based investigation and rely exclusively on another type of intellectual operation, no longer instrumental but intuitive. For this reason, only through the path of noesis, or direct vision, can one attain knowledge of the Good. This article analyzes the distinction between these two types of reasoning operations present in the Divided Line.
Keywords: hypothesis, divided line, good, Plato.
INTRODUCTION
In her commentary on the Republic, Julia Annas (1981, p. 6) says that Plato “wants to combine values of intellectual perfection, which require that life be dedicated to study, and values of practical activity, which require that the political world, which is in confusion, be improved.” This reading seems to gain strength if we take into account the testimony of Letter VII, which, as we know, narrates the three trips that Plato made to Sicily and, there, on the occasion of the last two, taking advantage of the philosophical aptitudes of Dionysius II, the philosopher would have tried to put into practice the government of the philosopher-king[1], which the Republic later realizes on a literary level.
In fact, in this work Plato brings together both philosophical and political depth. The dialogue addresses a multitude of themes, among which the main one is the problem of justice (δικαιοσύνη), introduced in Book I (330d). Along with the Symposium, Phaedo, and Phaedrus, the Republic belongs to the second phase of Plato's writing, forming part of the group of so-called middle dialogues, from the author's maturity. According to the relative chronology of the works, the Republic was written after the Meno and Phaedo and before the Parmenides, integrating the phase of elaboration of the theory of Forms[2].
After the writing of the Socratic dialogues, which characterizes the first phase of Plato's writing, Socrates continues to exert influence on the following dialogues, and this is because, to some extent, Plato considers that many of his ideas are the result of Socrates' influence (Annas, 1981, p. 04). The idea of the Good, for example, seems to be an unfolding of Socratic ethics (Stenzel, 1940, p. 28), which makes a lot of sense if we consider the early dialogues, which address moral issues more, as culminating in the Republic, where the theory of Forms is more fully developed, as shown by the image of the divided line.
The attentive reader cannot forget, therefore, that the death of Socrates is a political tragedy for Plato, who witnesses the master (a wise man) being condemned and killed by the Athenians themselves. In writing the Republic, Plato is concerned with the moral and political decadence of his people, largely caused by the relativization of social values. In this sense, one way to read the Republic is to take Plato as being concerned with Athenian moral skepticism, motivated, as is evident in many of his works, by the teachings of the Sophists. In this sense, all of Plato's efforts then go towards trying to find a safe moral parameter that serves as a guide to lead individuals to moral virtues. In the Republic, the realization of this political project is intrinsically rooted in the ontology and epistemology that dialogue offers, since it is from this theoretical and methodological basis that alternatives to a stable political life can be built. In this sense, philosophy is an important foundation that helps guarantee the order of the polis.
Books VI and VII, specifically, constitute what has been conventionally called the “Essay on the Good” (Pereira, 2014, p. XVI), but, in general, in this part of the Republic Plato dedicates himself to thinking about the preparation of the philosopher, in the sense of investigating “the manner and from which sciences and exercises there will be saviors of the constitution” (502c-d). Nourished with a good education, the guardians can, in the end, reach the highest knowledge, which is the understanding of the idea of the Good.
In this article we will show, therefore, that in the Republic Plato improves the application of the method of hypotheses (treated in the Meno and the Phaedo) and advances towards the idea of the good, which is beyond the hypothetical level (511b).
In 509d-511e, Socrates presents dialectics to us through the image of the divided line. Take, therefore, a line, divide it in half (establishing the sensible and intelligible planes) and divide each of these parts again so as to obtain four segments. In the first segment we find images, reflections and shadows and everything that resembles them. In the second, we find living beings, plants and objects of the same kind. The third, already in the intelligible, is the segment of mathematical and related objects. And, lastly, the segment where “the principle of everything” (511b) is found, the idea of the good (ἡ τοῦ ἀγαθοῦ ἰδέα). Socrates says that the soul, in traveling this path, goes through four stages: that of supposition (εἰκασία), of belief (πίστις), of discursive thought (διάνοια) until it reaches intelligence (νόησις), which comes through philosophy. In the end, it is seen that only he who reaches the supreme idea, the idea of the good, is dialectical. Having reached the summit, the philosopher then makes a descending path, recognizing, in the segments below, that the foundation of each thing, the absolute principle, is the good, which is the telos of dialectics.
There is a notable difference between what is stated in the Republic and what was presented in the Phaedo, with regard to the method of hypotheses. In the Republic, Plato makes clear the limitations of mathematics, which is only capable of reaching the third segment of the line, relating to dianoia. In this sense, goodness, beauty, and justice, in themselves, can only be known by the philosopher, because he is the only one capable of going beyond the hypothetical level. It is for this reason that Plato suggests in this dialogue that philosophers become kings, or philosopher-kings, a subject raised since the Gorgias (521d), developed in the Republic (473c-d), and further addressed in the Statesman (266e-267c). In the Republic, the philosopher is the most prepared to be the politician because he is the dialectician and, having known the idea of the good, is the only one capable of making the polis more just (435b; 540d).
The Method in the Image of the Divided Line
According to Robinson (1941, p. 69), method and intuition complement each other in the image of the divided line. This is perhaps the statement that best distinguishes the Republic from the Meno and the Phaedo, with regard to the process of philosophical investigation that presupposes the use of method. While the two dialogues that precede the Republic are restricted only to method, the Republic goes further and shows, in the divided line, that the method needs to be crowned by intuition (1941, p. 69). Contrary to what one might think, there is no contradiction in uniting method and intuition, since they complement each other and are necessary for good and orderly research, as suggested by Platonic orientation. Before, however, definitively entering into this subject, let us calmly analyze the image of the divided line and see how the method of hypotheses appears in this image.
In passage 503e, Socrates says that those who will become guardians, philosophers, “need to exercise themselves in many sciences, to see if they are capable of enduring higher studies.” Later, Adimantus asks Socrates if there is anything higher than justice, temperance, courage, and wisdom, virtues that have just been analyzed in the dialogue (504d). Socrates then says that the idea of the good is the highest study (ἡ τοῦ ἀγαθοῦ ἰδέα μέγιστον μάθημα, 505a). In 506d, with Glaucon's intervention, Socrates is pressured to give an “exposition on the good.” Unable, however, to accomplish this task, the master then resorts to a simile: “I want to explain to you what seems to me to be a child of the good and very similar to it” (506e). The result of this, as can be seen in the image of the sun (507b-509d)[3], is that Socrates will affirm that just as the sun is to the visible, the good is to the intelligible (508b-c), thus establishing the foundations of Plato's ontology in the Republic.
The image of the divided line (509d-511e) appears immediately after this account and presents a picture that situates the sensible and intelligible parts and, at the same time, establishes the levels of knowledge according to the divisions presented there. Socrates says: “Suppose then a line cut into two unequal parts; cut each of the segments again according to the same proportion, that of the visible kind and that of the intelligible kind” (509d). With the cuts made, we see four sections emerge, two relating to the sensible and two relating to the intelligible. So the scheme would look like this:
In the sensible world:
· Section 1: of shadows and reflections (509e-510a)
· Section 2: of living beings, plants and objects (510a)
In the intelligible world:
· Section 3: of mathematical objects (numbers, figures, etc. 510b; 510c)
· Section 4: of the “absolute principle” (510b)
In the final passage of the narrative (511d-e), Socrates reveals that these four sections are related to four operations of the soul, that is, if the soul, in a research process, turns to section 1, it obtains nothing more than supposition (εἰκασία), if it focuses on section 2, it operates only with belief (πίστις), in section 3, it operates with understanding (διάνοια) and, finally, in section 4, it operates with intelligence (νόησις). For now, we are interested in reflecting on sections 3 and 4, which is where Plato specifically deals with hypotheses, in the third segment, and the end of dialectics, in the last segment.
Hackforth (1942, p. 01), against the view that the third segment of the line is the place reserved for the “doctrine of intermediate mathematical objects” attributed to Plato by Aristotle[4], argues that what we find in this section 3 are in fact Forms, since what is there is located in the upper segment of the line. Hackforth's interpretation is that Plato constructs the divided line not only to illustrate the four stages of intelligence, as described above, but above all to more fully guide the discussion of the virtues (justice, temperance, courage, and wisdom) that comes earlier in the dialogue (504a; 504d). According to this reading, justice and the other virtues would only be understood by the guardian in the end if clarified by the idea of the Good, the absolute principle that is located in the last segment of the line. This interpretation presupposes that the objects found in the penultimate segment of the line are, first and foremost, Moral Forms (1942, p. 02)[5]. If we consider that Socrates is concerned not with a particular example of justice, but with justice itself, then taking virtues as Forms does not seem so absurd.
In 504c-d, Socrates says that for the guardian to reach the highest study, “he must go by the longest path.” This path is precisely the journey that the soul takes, with the help of dialectics, to the idea of the good. It is for this reason that the Forms that are located in the third
segment of the line are, so to speak, Forms “imperfectly known, because they are not yet known in the light of the idea of the good,” as Hackforth emphasizes (1942, p. 02, our translation).
Our interpretation, corroborating Hackforth's, is that the third segment of the line is the space already reserved for the Forms. Furthermore, we argue that the descending dialectical movement is entirely possible for this third segment, in the sense that such Forms have intelligible content and are therefore known by the dialectician.
The Problem of Mathematics
The mathematician's limitation lies in not being able to know the Forms as the dialectician knows them, because he has never reached the Form of the Good. We know that the method of hypotheses is certainly a methodological alternative inspired by mathematics, but its content, for the dialectician, is not limited to mathematical objects. Hence the philosopher deals with moral Forms (507b). The mathematician is restricted to investigating solely through hypotheses and does not have in mind other Forms, other than mathematical Forms. The reference to the method of geometers, in this section of the line, is justified more by the approximation that can be made to present the dialectical method, than as something that has the power to transcend and explain what a thing is in essence, as the philosopher does through dialectics[6]. The most that the mathematician can do is operate, on a descending scale, from the hypotheses of the Forms to sensible objects, as we saw in the Phaedo (101d-e).
According to Robinson (1941, p. 160), Plato's account of the use of hypotheses by geometers in the divided line makes it seem that only mathematics uses hypotheses, but dialectics also does. In describing the two segments of the intelligible, Socrates (510b) says: “the soul, using, as if they were images, the objects that were then imitated, is forced to investigate from hypotheses, unable to proceed to the principle, but to the conclusion.” This refers to the third segment. Regarding the fourth and final segment, he says: “whereas, in the other part, which leads to the absolute principle, it starts from the hypothesis, and, dispensing with the images that were in the other, makes the journey only with the aid of ideas.”
But there is a fundamental difference between dialectics and mathematics that needs to be highlighted. Although both make use of hypotheses, dialectics is not linked to sensibles, as mathematics is. Plato makes it clear that the starting point for mathematicians is the sensible world when they want to reach suprasensible realities (510d-e), while dialectics goes from hypotheses to the non-hypothetical principle.
Here, Plato seems to have seen that mathematicians themselves do not thoroughly investigate their practices, since they are not able to arrive at knowledge of the causes of what they investigate, nor are they able to justify the hypotheses they use (533b-c). According to Robinson (1941, p. 159), the proof that mathematicians are unaware of their starting points lies in their inability to provide any logos about such points.
In short, Plato criticizes mathematicians because they think they start from irrefutable certainties when, in fact, their certainties are nothing more than hypotheses, although they seem unaware of this. The divided line then shows that dialectics, unlike mathematics, always considers its premises only as hypotheses that, at any time, can undergo changes along the course of investigation, until one arrives at science (Robinson, 1941, p. 162), in other words, at dialectics.
Two Operations of Reasoning
Professor Cornford (1932, p. 37) observes that it is in the divided line that Plato first contrasts two modes of operation relating to the rational part of the soul: dianoia and noesis. For the author, this distinction does not mean that the Mathematical Forms (510d) are known only by dianoia and the Moral Forms (507b) are known by noesis. The Moral Forms are not a superior class. The difference between them is relative to their natures. The Mathematical Forms can be represented on the sensible plane, as is the case when someone draws a square or counts a certain number of things, whereas the Moral Forms are not susceptible to sensible representation, since it would not be possible to draw the idea of justice, for example, on any plane, although we know that such an idea is fundamental to social life.
Following this line of reasoning, it can be said that the Moral Forms are more difficult to know than the Mathematical Forms. Another point to be observed is that mathematical Forms can be objects of noesis, provided that the one who investigates them does so, through dialectics, in the light of the absolute principle (511d). In this sense, a mathematician could, through noesis, know the mathematical Forms (because they are intelligible), it would suffice that he looked at them from the idea of the good.
There is, then, a fundamental difference between these two types of knowledge. While noesis can be described as the ascending movement of intuition that goes from the hypothesis of moral Forms towards the absolute principle (the Form of the good), dianoia should be understood as the descending movement of deductive reasoning (Cornford, 1932, p. 43) that involves the articulation of premises and conclusion. This is a quite relevant distinction that we can establish between the Phaedo and the Republic. In Phaedo, we see that the dialectical movement starts from the hypothesis of the Forms and descends towards the conclusions, whereas in the Republic the dialectical movement takes an ascending direction, starting from the hypotheses to the principle that does not admit of hypothesis[7]. In other words, in the Republic the hypothesis continues to be an important component of dialectics, but the movement itself goes beyond the hypothetical method and “makes its way only with the aid of ideas” (510b), guided by intuition.
In general terms, we can say that the knowledge that comes from noesis presupposes intuition because it occurs in an “immediate act of vision,” in a “sudden leap,” while the knowledge that derives from dianoia is linked to a “continuous process” of investigation through discursive thought (Cornford, 1932, p. 48). Cornford (1932, p. 48) argues that the two movements of thought, dianoia and noesis, can be employed in the two upper segments of the divided line; that is, both the analytical power of noesis and the deductive reasoning process of dianoia can be directed towards mathematical or moral Forms. As for us, we are not certain that dianoia can reach the last segment of the line, although we agree that, through dialectics, it is possible to descend from the idea of the Good to the other ideas in the third segment. Our position is justified by the understanding that knowledge of the Form of the Good in the final stage of dialectics, through noesis, does not involve reasoning processes or logical articulations of premises and conclusions, but only direct vision, immediate intellection of the principle of everything.
The text itself, in Socrates' words, states that, through the power of dialectics, reasoning makes hypotheses a kind of support “to reach that which admits no hypotheses, which is the principle of everything, reaching which it descends, focusing on all the consequences that follow from it, until it arrives at the conclusion” (511b-c). In this sense, there is a substantial difference between the mental state of the mathematician and the mental state of the philosopher, namely, the mind of the mathematician does not have noesis, but dianoia (Cornford, 1932, p. 50). This is why, for Plato, only the philosopher has a “perfectly clear vision” (nous) of the principle, since he is capable of an “intuitive apprehension” of the idea of the good (Cornford, 1932, p. 51)[8].
Development and unity of dialectics
The notion of intuition, in Plato, is associated with mathematics. According to Stenzel (1940, p. 38), intuition is the representation of the universal in the particular. According to this interpretation, the mathematical model of reaching the universal from the abstraction of particulars was used by philosophy even in the theories of ancient thinkers. In the specific case of Plato's dialogues, Stenzel believes that the notion of idea as the essence of particulars is derived from the mathematical model that predicts that, to some extent, the universal is already present in particular cases. In this sense, even if it were not possible to arrive at conclusive definitions about beings, an intuitive vision of the universal would be possible for philosophy. It is from this, according to Stenzel, that Plato constructed the doctrine of ideas.
We must observe, however, that the sense of intuition that we are exploring in this text is specifically associated with the intuitive apprehension of the absolute principle, as happens, through the dialectical movement, in the last segment of the divided line. We believe that, in Phaedo, for example, or in the third segment of the line, where we have already noted the presence of the Forms, what we have is deductive reasoning, whereas intuition is only possible when one goes beyond hypotheses, when the philosopher makes use of noesis (Kahn, 1998, p. 320).
Regarding the application of the method of hypotheses, we must remember that, although inspired by mathematics, the Meno relates it to the investigation that revolves around virtue. There, the method receives criticism from Socrates for not being able to define what a thing is[9]. In the Phaedo, the method of hypotheses, which is also influenced by geometers, has much more consistency, since it is anchored in the doctrine of Forms (the strongest logos), but dialectics does not appear fully developed there, even if it is assumed that the dialogue already makes some allusion to the Good as a principle (99c). It is in the Republic that we clearly see that the starting point of the hypotheses is mathematics, as described in the third segment of the divided line. However, this knowledge that comes from the hypothetical method of mathematicians can be surpassed by the power of dialectics (διαλέγεσθαι δυνάμει, 511b), because dialectical science (διαλέγεσθαι ἐπιστήμης, 511c) is the only one capable of arriving at the principle of everything (παντὸς ἀρχὴν, 511b)[10].
According to Kahn (1998, p. 320), Plato has four methods of hypotheses, which appear respectively in the Meno, the Phaedo, the Republic, and the Parmenides, but this does not mean that there is a substantial change in Plato's thought, as if there were four. Distinct theories, on the contrary, what exists are different aspects of a single theory. The three intellectual operations involved in the method of hypotheses have the same characteristics in these dialogues: “postulating a supposition, deriving results that fit, and justifying, removing, or otherwise 'accounting' for the supposition.[11]“ Kahn's thesis is that these operations, with all their distinctions and connections, target intelligible realities, the Forms, which confers unity to the method of hypotheses as a component of dialectics. This interpretation values what Plato says in the Republic: that “whoever is capable of having an overview is the dialectician” (537c). If this is the case, even if the keyword for dialectic in the middle dialogues (Meno, Phaedo, Republic and Parmenides), as Robinson (1941, p. 74) says, is “hypothesis” and the keyword for the later dialogues (Phaedrus, Sophist, Statesman and Philebus) is “division”, it is possible to see the dialectical method as a unity, if we consider that Plato always uses it to address suprasensible realities.
For us, this reading perspective that takes into account the unity of dialectics in Plato's thought in no way excludes the idea that Plato refined the use of the method throughout the dialogues.
In a developmental approach, Sayre (2017, p. 83) draws attention to the fact that, in the Republic, the method of hypotheses and the method of gathering and division are united. As we saw earlier, dialectics is presented in the divided line as the method that makes hypotheses “a kind of steps and points of support, to go up to that which does not admit hypotheses” (511b), the idea of the good. But there are two passages in the dialogue that mention the method of gathering and division, and tradition does not usually highlight them. The first instance occurs in Book V, in 454a, where Socrates emphasizes that the dialectician must be able to divide according to the Forms (κατ᾽ εἴδη διαιρούμενοι), this orientation aims to mark an opposition between dialectics and eristic antilogy[12]. The second passage is in Book VII, in 531d, where Socrates tells Glaucon that the method they are employing in all sciences must be able to lead them to “what is common and related among them and demonstrate their reciprocal affinities,” so that the work will not be in vain. Further on, Socrates says that “when someone tries, through dialectics, without using the senses and only by reason, to reach the essence of each thing, and does not give up before having apprehended only by intelligence the essence of the good, he reaches the limits of the intelligible” (532b)[13]. As Dixsaut (2013, p. 91) will affirm, dialectics in the Republic has a telos: the good.
CONCLUSION
This article sought to demonstrate that there are two distinct types of reasoning operations in the divided line of Plato's Republic. This occurs because their objects are also distinct. When we need to deal with logical, procedural, or instrumental thought, that is, the kind that requires articulations of premises to arrive at a certain conclusion, we make, so to speak, use of dianoia. And when we need to arrive at the absolute principle, at the intelligible unity of the good, as described in the fourth segment of the line, then we necessarily have to resort to noesis, that type of intelligence that occurs through an act of (overall) vision, direct and intuitive. It is for this reason that the method of hypotheses, typical of dianoia, of instrumental thought, is not capable of advancing the investigation towards the principle of everything. However, in the divided line Plato shows us that dialectics can make use of these two operations of reasoning, because, after the sudden vision of the good, the philosopher can descend again to the zone of dianoia and use this methodological instrument that is the hypothesis, to arrive at certain conclusions. But, this time, knowing the absolute foundation: the Form of the good.
After the Republic, Plato changes the instruments of dialectics. The reason for this is that the objects of investigation in the later works no longer require mathematics as a model. In dialogues such as Phaedrus, Sophist, and Statesman, the author approaches the natural sciences and begins to use the method of gathering and dividing in his investigations.
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Acknowledgements
National Council for Scientific and Technological Development, CNPq.
Weriquison Simer Corbani
Researcher in productivity at IFES. He holds a degree in Philosophy from the Federal University of Espírito Santo (2010), a master's degree in Philosophy from the Federal University of Espírito Santo (2013), and a doctorate in Philosophy from the Federal University of Espírito Santo (2023). He is currently a professor of Philosophy at the Federal Institute of Espírito Santo (IFES). He is the founder of the research group “PHRÓNESIS - Ethics and Political Theory” (CNPq). He participates in the Research Ethics Committee with Human Beings (CEP) of IFES. He is a member of the Brazilian Society for Classical Studies (SBEC) and the Center for Hellenic Studies (Areté). His doctoral thesis, “Is Dialectics a Political Science? - A political-philosophical study of Plato's Statesman,” was a finalist for the ANPOF 2024 Award, ranking among the four best theses in the Philosophy area. He is a researcher in the field of Philosophy, with an emphasis on the History of Ancient Philosophy, Ethics, and Political Philosophy.
The texts in this article were reviewed by third parties and submitted for validation by the author(s) before publication.