theory and practice in plato’s        Psychology of mathematics education

Stephen R. Campbell

Simon Fraser University

Plato made contributions of the first rank to education in areas such as administration, cognitive theory, curriculum development, teaching and pedagogy. I do not provide a detailed overview of these contributions here. Rather, I focus on some of his theoretical and practical contributions to the psychology of mathematics education, especially his three similes (the Sun, the Divided Line, and the Cave) considered with regard to mathematical cognition, and his mathematical curriculum. I conclude with some crucial questions regarding the implications of Plato’s work for mathematics education.

Introduction

Plato (429-347 B.C.E.) is arguably one of the most important educational theorists and curriculum developers in Western history. Moreover, his contribution to the theory and practice of mathematics education has had a profound impact over the ages.

Plato’s educational interests and accomplishments were clearly founded on mathematics education. The entrance to the Academy he established in Athens (instituted from 387 B.C.E. to 526 A.C.E.), famously announced: “let no one unversed in geometry enter here.” In his classic tome, Republic, the mathematical sciences (arithmetic, geometry, astronomy, and harmonics) formed the foundation of Plato’s curriculum. Enduring well into the Middle Ages, these four disciplines comprised the quadrivium, which coupled with the trivium (grammar, logic, and rhetoric), constitute the so-called seven liberal arts.

Of more contemporary interest however, and my main purpose here, is to explicate Plato’s model of mathematical cognition and learning. Considering his mathematical curriculum in this light, Plato may readily be seen as having put theory into practice on a scale unprecedented, before or since, in the history of mathematics education.

Plato’s Cognitive psychology

Overall, Plato’s cognitive psychology appears more structural and descriptive in nature than developmental. He was, so it is said, more interested in timeless entities and truths than he was in transient processes and beliefs. Interesting parallels emerge with modern psychology, as we shall see, when considering the temporal component of Plato’s archetypal model of cognition.

The heart of Plato’s cognitive psychology is to be found in the Republic (1945/ca.366 B.C.E.).[1] There, he offers a basic division of psyche (and, by analogy, polis and cosmos) into rational, spirited, and appetitive dispositions (ibid., 434d–441c). In addition to this tripartite division of interrelated faculties, Plato’s psychology is largely bound up with explicating three main similes, or allegories: 1) the Sun, 507a–509b; 2) the Divided Line, 509d–511d; and 3) the Cave, 514a–517a. These similes are informative in and of themselves, but they also serve to inform each other in important ways as well. Of course, scholars have interpreted these similes in many different ways. The following is offered from a perspective of research in the psychology of mathematics education.

The Sun

Plato uses the sun as a similitude for his highest transcendent object, or Form:[2] the good. The epistemological correlate of this ontology is knowledge of the good. That knowledge is based on a logical distinction of Parmenides (Campbell, 1999) between things in a state of eternal being, and things in a transient state of becoming. The questions arise: What is the good? And how is knowledge of the good to be sought? Plato’s simile of the sun is meant to provide insight to the former of these two questions. As we shall consider in more detail below, Plato’s mathematically oriented curriculum is designed and intended in large part as an educational response to the latter question.

Evading his interlocutor Glaucon’s repeated requests for a definition of the good, Socrates, as Plato’s designated protagonist in the dialogue, succumbs to offering a provisional ‘token’ account by likening the good to the sun. Before so doing, however, Socrates carefully reminds Glaucon “of the distinction we drew… between the multiplicity of things that we call good… and… goodness itself, …a single Form or real essence, as we call it… the many things, we say, can be seen, but are not objects of rational thought; whereas the Forms are objects of thought, but invisible” (507b).

Socrates’s qualification here can serve to remind us, for all intents and purposes, that Plato’s Forms are what we typically feel more comfortable referring to today as concepts. For Plato, though, it is also important to remember that the transcendent Forms are eternal, not transient, and any knowledge thereof (which Plato, writing in the literary guise of Socrates, ironically denies having)[3] must be apodictic, i.e., indubitably certain.

Socrates identifies eyesight, somewhat dubiously, as unique in requiring mediation, qua the light of the sun, in order to perceive its objects. Similarly, he argues, when the intellect is turned toward the Forms, they in turn must be “irradiated by truth and reality” in order for the soul to gain any insight regarding them. Moreover, just as the sun makes things visible, Socrates also credits the sun with bringing all things visible into existence. In correlating the sun (with regard to the visible objects of perception) with the good (with regard to the invisible objects of intellect, e.g., knowledge and being), Plato exalts the good not only as the highest Form, but that one singular Form which stands above, accounts for, and gives rise to the Forms of knowledge and being (508d-509a).

Naturally, there are other kinds of light, but one sees best in sunlight, and that is certainly good in a way. Similarly, there may be lesser kinds of truth and reality, but presumably they would all fall significantly well short of the universal truths and timeless realities that are rightfully of concern to intellection emanating from the good. If all this sounds somewhat esoteric, a case in point would be that a geometric line conceived by intellect is considered as different in kind, and as transcending any facsimile perceptible to the eye. But how might intellect be involved or implicated in conceiving such things?

The Divided Line

The simile of the Sun establishes by analogy that there are two separate realms accessible to human cognition: A transient, changing realm perceptible to the senses, and a timeless, eternal realm that is conceivable to the intellect. The simile of the Divided Line adds further granularity and nuance to this picture. Indeed, this simile, I would like to suggest, can be seen as the first substantive cognitive model in the history of Western thought.

World of Appearances                        Intelligible World

                 A                  B                          C                                    D

 

Figure 1.

The simile of the Sun provides the basis for distinguishing between objects in the visible world perceptible by the senses and objects of an intelligible world accessible by thought. With this distinction in mind, Socrates invites his audience to “… take a line divided into two unequal parts, one to represent the visible order, the other the intelligible; and divide each part again in the same proportion, symbolizing degrees of comparative clearness or obscurity” (509d) (Figure 1). Corresponding to the divided line in Figure 1 is Table 1, the latter indicating this “comparative clearness or obscurity” by the opacity of the cells. Socrates goes on to explicate the resultant four sections of the divided line as indicated by the added content (after Cornford, 1945), distinguishing between different states of mind and their respective objects placed in Table 1.

	Objects	States of Mind
D	Forms	Intelligence (noesis) or Knowledge (episteme)
C	Mathematical Objects	Thinking (dianoia)
B	Visible Things	Belief (pistis)
A	Images	Imagining (eikasia)

Table 1 (after Cornford, 1945).

Socrates is quite clear in this simile as to the two distinct kinds of objects that the “lower” part of the divided line is meant to designate. Section B designates “visible things,” the common objects of perception, whereas section A quite literally designates shadows, reflections, and similar kinds of indirect representations of the common objects of perception from section B. With regard to our sense of hearing, section A would presumably include echoes of real sources, the latter of which would fall under section B.

What marks the difference regarding the two kinds of objects designated by the “upper” part of the line is, prima facie, less evident. Once again, Socrates draws an analogy with the perceptual world of appearances. He considers objects of section C with respect to the objects of section B analogously to the way that objects of section B relate to objects of section A. That objects of C can be designated as mathematical can be seen, for example, by noting that the image of a pure geometrical line in the mind (taken as an object of C) is to a visual drawing of that line (an object of B) as the visual drawing of that line would be to a reflection of that drawing in a mirror (which would result in an object falling under section A). But can objects falling under C solely be mathematical? Objects of D can figuratively be taken to relate to objects of C in an analogous fashion to the way in which objects of B relate to objects of A, but as the objects of section D are the Forms, how are they to be actually realised or conceived of in the mind?

In summary, the various proportionality relations of the various ratios of sections of the divided line with respect to each other are: [4]

A+B : C+D :: A : B :: C : D :: C : B

Cornford (1945) points out that although Socrates refers to the “lower part,” i.e., A+B, in the simile of the Divided Line as representing the “visible order” of things (as per the simile of the Sun), elsewhere in the text, this world of appearances is taken more generally to include Doxa, i.e., unsubstantiated opinions and conventional beliefs. Consider, furthermore, that section C might also include non-mathematical objects, i.e., other paradigmatic images of particular objects from B, e.g., an ideal image of a person, and what of the differences between belief and perception with respect to objects of B? Moreover, what are we to make of the difference between noesis and episteme? Cornford’s portrayal of this model regarding “States of Mind” seems somewhat muddled in this regard, however, distinguishing intuitive from propositional modalities of cognition can restore a semblance of order in Plato’s model. This distinction will only be described (see Table 2) and not justified here.

	Objects	States	of  Mind
		intuitive	propositional
D	Forms	Intelligence (noesis)	Knowledge (episteme)
C	Mental Objects	Ideal imagining	Thinking (dianoia)
B	Actual Things	Normal perception	Belief (pistis)
A	Illusions of Things	Actual imagining	Imagining (eikasia)

Table 2.

In this revised consideration of Plato’s cognitive model based on his simile of the Divided Line, with the exception of the Forms, the objects under consideration in the other sections seem more reasonable. The objects of section C are still exemplified by mathematical objects, but are not restricted to them. They include anything that is an idealisation of some actual kind of thing, like geometric lines, chariots, or political systems. The category of objects in section B is broadened to include anything actual that is perceived, not just seen. Objects under A are now specified as illusions.

Briefly, with regard to intuitive states of mind, differences between idealised images and illusory images can be defined by noting that the former are ‘internal’ mental objects and the latter are ‘external’ reflections of actual objects. Whereas the internal images are in some sense idealised, and thereby simplified, the external images are typically distorted, and thereby rendered more complex. As to what exactly would constitute an intellectual intuition of the Forms, better indicated here by noesis than dianoia, remains a problem.[5]

Distinguishing propositional states of mind, ostensibly a more linguistic kind of modality, helps account for the inclusion of Doxa, i.e., aforementioned unsubstantiated opinions and common beliefs, in the “world of appearances,” i.e., with regard to the lower part of the Divided Line. Propositional imagining could include anything ranging from hypothetical conjectures regarding the state of the world to delusional beliefs, whereas common beliefs could be restricted to statements regarding matters of fact. In both cases, due to the empirical, and thus temporal and contingent, orientation of these propositions, they are rendered inappropriate for inclusion in the higher part of the line. On the other hand, in the higher part of the line, Plato does admit true opinion, as a kind of knowing, i.e., when an opinion, unbeknownst to the one holding it, happens to hold true. Any and all knowledge in this propositional modality of mind can be justified on the basis of rational criteria, e.g., consistency, and presumably may ultimately be based on noesis. There is much more to be said about this model than will be treated here. We will, however, revisit some aspects of it in what follows.

The Cave

Plato’s simile of the Cave charts a course between theoretical matters of cognitive psychology toward practical challenges involved in educating learners toward the good light of reason and the eternal forms of being and truth. More specifically, the simile of the cave incorporates the similes of the Sun and the Divided Line, while providing a structure and supporting rationale for Plato’s mathematical curriculum.

In the cave, prisoners are bound and chained, constrained in their experience to mere shadows projected upon the wall in front of them and the sound of echoes around them. Their experience is restricted to the objects of illusion, the lowest section of the line. From this, they can only speculate on the way things really are. Behind them is a wall, and the happenings behind that wall are likened to a shadow puppet show, conducted by conjurers, backlit by a fire, projecting those images the unwitting prisoners are privy to. The objects projecting the shadows can be taken as stand-ins for the objects of section B, and the conjurers themselves are in the distinct position of knowing that there is a deeper reality the puppeteer’s objects represent that stands in relation to those objects, analogous to the way, as we have seen above, the objects of C stand in relation to the objects of B. Those real objects the puppets represent, however, are only to be found outside the cave. Plato, the real voice behind his literary mask of Socrates, has us imagine that one of the prisoners is unchained, compelled upwards, out of the cave into the blinding light of day.

This forced process is likened to subjecting a student to the higher part of the line without adequate preparation: “… would he not suffer pain and vexation at such treatment, and, when he had come out into the light, find his eyes so full of its radiance that he could not see a single one of the things that he was now told were real?” (515e-516a). Here we have a clear allusion to a developmental path for navigating between theory and practice. The prisoner, qua student, should not be fooled, as a case in point, into thinking that mathematics is only about manipulatives. Rather, there is something beyond the objects of section B that relate to those objects analogous to the way a puppet projects a shadow. But for a student to enter the realm of mental objects, they must be led out of the cave of appearances, but slowly and at night, not rushed too quickly into the full light of day—the light of day being “dialectic,” the reasoning underlying proof (Campbell, 2002).

Plato has a similar warning for a now enlightened teacher who would venture out of the light of day too quickly back into the darkness of the cave. Sitting back again amongst his fellow prisoners, “[h]e might be required once more to deliver his opinion on those shadows, in competition with the prisoners who had never been released, while his eyesight was still dim and unsteady… They would laugh at him and say that he had gone up only to come back with his sight ruined” (516e-517a). Here Plato provides excellent advice to the aspiring educator, i.e., the need to address learners from where they are at, not from the lofty realms into which the aspirant has traveled. That is to say, the simile of the Cave reveals that Plato’s simile of the Divided Line is not just a philosophical theory regarding the nature of things, but also, an archetypal developmental stage theory that has guided and informed educational practice in mathematics education for millennia.

the purpose of Plato’s mathematical Curriculum

For Plato, the purpose of education, and mathematics education especially, is figuratively to turn the learner’s soul from the dark shadows of the cave to the light of reason. But this distinction between night and day, between the realms of appearance and intellect, is not to be determined as if “… in a children’s game, by spinning a shell” (521c). Much more is at stake. The more that the psyche and, so too, the polis, resonate with the natural harmonies of the cosmos, the greater the good to be realised. Pythagorean that he was, anticipating Newton by over two millennia, Plato was convinced that the cosmos operated in accord with “a law of number,” and that it was within human potential to discover that law. It is no coincidence that Plato called his esteemed colleagues in the Academy to task to account for the “apparently inconsistent motions of the planets…, by compositions of invariantly constant circular motions” (Vlastos, 1988, p. 362).

For Plato, however, unlike today, the study of the quadrivium of mathematical sciences, to the exclusion of anything else, was a ten-year process of higher education for citizens (i.e., the guardian class) from the age of twenty to the age of thirty. This exclusive focus on the topics of arithmetic, geometry, astronomy, and harmonics served as “protreptic and proleptic instruments, positioning the [student] dispositionally and providing hints for the work of completing the direction of thought [toward the good] by attending to ‘the things themselves’” (Wood, 1991, p. 525).

Implications for mathematics education

Plato’s theories and practices in mathematics education have exerted an influence unprecedented and unparalleled in the history of our field. One may take issue with his ontology of transcendent Forms, but one may be unwise to discount it too readily. The issues involved in explicating these ideas, in Platonic terms or otherwise, remain at the forefront of research in the philosophy of mathematics. For instance, in considering an infinite line as opposed to a finite line segment as a mental object, at what point and in what ways does imagination fail? At what point must we resort to propositional definitions and proofs? What happens between the Aristotelean notion of a potential infinity defined by n+1 for any natural number n, and the Platonic notion of an actual infinity defined by the set of natural numbers N? Questions such as these take on more contemporary relevance and immediacy for research in the psychology of mathematics education when considered from a cognitive rather than an ontological perspective.

References:

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Campbell, S. R. (2001) Number theory and the transition between arithmetic and algebra: Connecting history and psychology. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.) Proceedings of the 12th International Conference in Mathematical Instruction Study Group on The Future of the Teaching and Learning of Algebra (vol. 1; pp. 147-154), Melbourne, Australia: University of Melbourne.

Campbell, S. R. (1999). The problem of unity and the emergence of physics, mathematics, and logic in ancient Greek thought. In L. Lentz, & I. Winchester (Eds.) Proceedings of the 4th International History and Philosophy of Science and Science Teaching Conference (pp. 143-152). Calgary, Canada: University of Calgary.

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Snell, B. (1982/1953). The discovery of the mind in Greek philosophy and literature. 21(2), 132-144. New York: Dover Publications, Inc.

Vlastos, R. E. (1988). Elenchus and mathematics: A turning-point in Plato’s philosophical development. The American Journal of Philology, 109(3), 362-396.

Wood, G. (1991). Plato’s line revisited: The pedagogy of complete reflection. Review of Metaphysics, 44, 525-547.