Plato's Allegory of the Cave: Part Two

As mentioned in part one, Plato, through the character of Socrates, is looking for the knowledge which can draw the soul from becoming into being. Socrates reminds Glaucon that the young men who are to learn this knowledge will be warrior athletes in order to govern the State. Therefore, this new knowledge must possess an additional quality -“usefulness in war”. Gymnastics, Socrates claims, presides over the “growth and decay of the body”, and therefore regards the knowledge of “generation and corruption”. However, both Glaucon and Socrates agree that this is not the knowledge they are looking for.

Socrates then digresses to music, asking Glaucon for his opinion on what he refers to as the other part of the education system. Glaucon calls music a “counterpart of gymnastic”, and states that is what makes the guardians “harmonious” and “rhythmical” but it cannot give them Science. Thus, there is also nothing in the knowledge of music which “tends to the good” of what Socrates is seeking.

The arts is also excluded. Socrates asks what knowledge remains of their “desired nature” now that gymnastics, music and the arts have been excluded. He states, “There may be nothing left of our special subjects; and then we shall have to take something which is not special, but of universal application”. Therefore, they aren’t look for a distinctive feature in gymnastics, music and art, but a feature which they all share. The something that Socrates refers to is “a something which all arts and sciences and intelligences use in common, and which everyone first has to learn among the elements of education”.

Arithmetic

This is when Plato’s philosophy starts to develop regarding education and what he believes to be the ideal branches of study. The common factor of all the arts and sciences, Socrates states, is “the little matter of distinguishing one, two and three, in a word, number and calculation”. Glaucon agrees that all of these types of knowledge partake in this mental activity; the basics of mathematics. The art of war also partakes, and Socrates uses an example situation of Palamedes, a Trojan hero, who “proves Agamemnon ridiculously unfit to be a general”. He details how:

“[Palamedes] invented the number, and had numbered the ships and set in array the ranks of the army at Troy, which implies that they have never been numbered before, and Agamemnon must be supposed literally to have been incapable of counting his own feet - how could he if he was ignorant of number?... And what sort of general must he have been?”

Glaucon remarks “a strange one”. Although Agamemnon is the general, he lacks mathematical skill which makes him a subordinate to Palamedes regarding intelligence. Socrates and Glaucon agree that a warrior must therefore have knowledge of the arithmetic.

Socrates asks Glaucon whether he shares the same notion as him on this study. His notion is that:

“It appears to me to be a study of the kind which are seeking, and which leads naturally to reflection, but never to have been rightfully used; for the true use of it is simply to draw the soul towards being”

The term reflection here mirrors Plato’s previous use of the term ‘contemplation’. He believes the end of the prisoner’s journey to be deep thought about the sun and the visible world. This knowledge, which he attempting for the future guardians to obtain, is a study which results in reflection. Parmedes, using the arithmetical part of his brain, has made a mathematical breakthrough. He achieved this through rational thinking.

Picture every single human’s mind being encompassed by a veil, and as they turn the innate knowledge in their brain towards the right direction, this knowledge is being excelled. The knowledge then pierces through the veil, pricking the world of philosophy, and opens the human’s mind to the true knowledge, until this veil eventually disappears. Their soul is now contemplating the world around them.

Glaucon asks for a further explanation of the meaning of Socrates’s notion. He begins by discussing objects of sense. This come in two kinds:

• Objects of sense which do not invite thought. The sense is an adequate judge of them.

• Objects which the sense is untrustworthy and requires further investigation.

Glaucon believes that Socrates is referring to “the manner in which senses are imposed upon by distance, and by painting in light and shade”. Socrates denies this:

“When speaking of uninviting objects, I mean those which do not pass from one sensation to the opposite; inviting objects are those which do, in this latter case the sense coming upon the object, whether at a distance or near, gives no more vivid idea of anything in particular than it’s opposite”

To explain this, Socrates gives an example by holding up three fingers, which Glaucon agrees are fingers and seem close. Socrates explains that the finger is always assumed to be a finger, whether “white or black, thick or thin”. Man is not compelled to ask the question what is a finger, “for sight never intimates to the mind that a finger is other than a finger”. “There is nothing here that invites or excites intelligence”, he says. The finger is an object of sense for it does not provoke any further thought and the senses can be trusted.

The explanation continues. “Is this equally true of the greatness and smallness of fingers?” he asks, “Can sight adequately perceive them?... And so of the other senses; do they give perfect intimations of such matters?” When we see an object from a distance, it can appear smaller than it actually is. Our eyes however, see the object and pass on to our mind, and we can usually deduce that if the object was closer to us, it may appear bigger, for example a ship sailing on the horizon.

“... the sense which is concerned with the quality of hardness”, Socrates explains, “is necessarily concerned with the quality of softness, and only intimates to the soul that the same thing is to be felt both hard and soft… and must not the soul be perplexed at this intimation which the sense gives of a hard which is also soft… what is the meaning of light and heavy, if that which is light is also heavy, and that which is heavy, light?” Glaucon replies that he feels the intimations which the soul receives are “very curious”. In order to perceive an object as soft, we must also be aware of the sensation of hard to determine which is soft.

Socrates explains that the complexities of the soul require the aid of calculation and intelligence so that the soul can decipher whether objects in front of them are “one or two”.

“Is each of them one and different?... And if each is one, and both are two, she [the soul] will conceive the two as in a state of division, for if they were undivided they could only be conceived of as one?”

Socrates is discussing basic mathematical concepts. Imagine two pencils, side by side. Together they are one, but if you separate the pencils, you have two. The mind applies this simple calculation to multiple concepts, including two opposing sensations which can be compared and divided, such as heavy and light. But let’s stay on the path of physical objects that can be observed by the eye, such as the pencils, and apply the concept of small and great to them. What the eye and mind see can differ with regards to intellect and rational thinking.

“The eye certainly did see small and great, but only in a confused manner; they were not distinguished… Whereas the thinking mind looks at small and great as separate and not confused”

Therefore, the eye sees an immediate jumble of objects, but this image passes from the vision through to the mind, and then these objects undergo the mental calculation of division. Imagine a table full of different objects, and in your head these objects are counted, separated and thus categorized. On this table are pencils, some are bigger than others. The eye sees the pencils, but the mind recognizes that one is small and the other is great.

This is the distinction between the visible and the intelligible.

And this is what Socrates was referring to when he mentioned objects which invite thought and those which do not; “those which are simultaneous with opposite impressions invite thought; those which are not simultaneous do not”. But “to which class”, Socrates questions, “Do unity and number belong?”

“If simple unity could be adequately perceived by the sight or by any other sense, then, as we are saying in the case of the finger, there would be nothing to attract towards being; but when there is contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks, “what is absolute unity?”. This is the way in which the study of the one has the power of drawing and converting the mind to the contemplation of true being”

This again returns us to the importance of contemplation. Applying numerical values to objects in the visible world evokes thought in the mind, and the calculations carried out expand knowledge on the physical world around us. When we label objects as one, two, three and so forth, we are opening a huge rational part of our mind which not only leads to contemplation, but influences a variety of aspects in the world around us including communication and language. Remember the example Socrates gives of the leader Agamemnon being unable to count his own feet? Surely arithmetic are a vital skill for any warrior, especially the governor of the State.

Both Glaucon and Socrates agree that the knowledge of arithmetic leads us towards the truth. The knowledge that they are seeking, as previously stated, must have more than one type of usefulness. Arithmetic has both a use in philosophy (contemplation) and military, “for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also because he has to rise out of the sea of change and lay hold of true being, and therefore must be an arithmetician”.

“Our guardian is both warrior and philosopher”

Socrates insists that this is the knowledge that the principal men of the State should endeavour to learn, and they should aim to persuade these men to study arithmetic until “they see the nature of numbers with the mind only… for the sake of their military use, and of the soul herself… this will be the easiest way for her to pass from becoming to truth and being”.

Arithmetic holds a colossal importance to Plato’s idea of the perfect guardian. He talks very highly of abstract numbers, and how they compel the soul to reason, “rebelling against the introduction of visible or intangible objects”. Numbers have unity, he explains, “each unit is equal, invariable and indivisible”, and these numbers also can only be obtained rationally, as according to Glaucon. They agree that this knowledge is wholly necessary, for it exercises intelligence to pursue pure truth which can only be attained in the corners of the mind.

“Have you further observed”, Socrates asks, “That those who have a natural talent for calculation are generally quick at every other kind of knowledge?” Thus, this knowledge must not be forgotten, and instead nurtured and encouraged.

Geometry

The second type of necessary knowledge Socrates describes is arithmetic’s “kindred science”, which Glaucon rightly suggests is Geometry.

“We are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of the army… it will make all the difference whether a general is or is not a geometrician” - Glaucon

Socrates explains that they are discussing advanced geometry, the part which will aid the soul to “turn her gaze towards the place, where is the full perfection of being, which she ought, by all means, to behold”. Therefore, they are discussing similar knowledge to that of arithmetic which will ultimately achieve the idea of the good.

“If geometry compels us to view being, it concerns us, if becoming only, it does not concern us?... Yet anybody who has the least acquaintance with geometry will not deny that such a conception of science is in flat contradiction to the ordinary language of geometricians?” Glaucon questions how, for he feels that geometricians always confuse “the necessities of geometry with those of daily life”. So how should geometricians be speaking, and what is the correct manner to approach the study? Knowledge, must be the real object of the whole science.

Socrates claims that “a further admission must be made”. The knowledge which geometry aims at achieving must be not perish nor be transient, and is there knowledge of the eternal, just like numbers.

“Geometry will draw the soul towards truth, and create the spirit of philosophy”

All inhabitants of the city should learn geometry, for the advantages of learning this study have great impacts, such as seen in the example of military use. “Anyone who has studied geometry is infinitely quicker of apprehension than one who has not”, Socrates says.

Astronomy

The third branch of study to teach the youth, with geometry as the second and arithmetic as the first, is astronomy.

However, Socrates suddenly retracts this claim, placing astronomy as the fourth. Astronomy’s third place is occupied with “the ludicrous state of solid geometry”, “concerned with cubes and dimensions of depth”, i.e, the third dimension. He has divided the study of geometry into two parts, the first which revolves around the question of position, shape and size, and the second of 3D shapes.

The order of study follows:

1. Arithmetic (rational numbers)

2. Plane Geometry (properties of space/ position)

3. “Solid” Geometry (third dimension)

4. Astronomy (the motion of these solids)

Socrates believes that little is known of the third study because “no government patronises them” and “students cannot learn them unless they have a director, but then a director can hardly be found”. But if a director was found, he feels that the students would not listen unless the State “gave honour” to the study. This, combined with the study’s ‘natural charm’, is the only way Socrates feels this vital branch of education will ever come to light.

Glaucon and Socrates then discuss astronomy. Glaucon claims that the study “compels the soul to look upwards and leads us from this world another”. Socrates, however, disagrees. He believes that astronomy is the opposite; looking downwards.

“If a person were to throw his head back and study the fretted ceiling, you would still think that his mind was the percipient and not his eyes… but in my opinion, that knowledge only which is of being and of the unseen can make the soul look upwards, and whether a man gapes at the heavens or blinks on the ground… I would deny that he can learn, for nothing of that sort is matter of science; his soul is looking downwards, not upwards”

Glaucon requires a further explanation. Socrates explains:

“The starry heaven which we behold is wrought upon a wide ground, and therefore, although the fairest and most perfect of visible things, must necessarily be deemed inferior for the true notions of absolute swiftness and absolute slowness, which are relative to each other, and carry with them what is continued in them, in the true number and in every true figure… These are to be apprehended to by reason and intelligence, not by sight”

Therefore, gazing up at the stars and debating over the heavens distracts the eye from true knowledge, which can only be obtained by the mind. We must focus instead on what is on the ground below, and what is within us, rather than dwell on assumptions about the metaphysical world. When contemplating the stars, we are turning the eye of the soul in the wrong direction.

However, we should still appreciate the stars. “Any geometrician who saw them would appreciate their workmanship, but he would never dream of thinking than in them he could find the truth”. Socrates believes that any astronomer would feel the same, and that upon observing the stars and their motions, “he will never imagine… things that are material and visible can also be eternal and subject to no deviation. ” Planets, like all visible objects, are subject to change unlike the timeless knowledge and unity of numbers. Thus, we should leave the heavens alone for the “gift of natural reason to be of any real use”.

Socrates then address the study of motion. Motion has two forms: one which is obvious to anybody, and one which “may be left to wiser persons”. These are counterparts of each other. “The second… would seem relatively to the ears to be what the first is to the eyes; for I conceive that as the eyes are designed to look up at the stars, so are the ears to hear their harmonious motions; and these are sister sciences”. However, he states this study, although worth learning, is “laborious” and that “we must not lose sight of our higher objective”.

Glaucon asks what this higher objective is. Socrates responds, “A perfection which all knowledge ought to reach and which our pupils ought also to attain”. He claims that the science of harmony is similar to that of astronomy, for the teachers of harmony compare the sounds and consonances which are heard”. Although, their labour is “in vain”, for they argue over notes, and the difference between sounds, but they never reach the natural harmonies of number, or reflect why some numbers are harmonious and others are not. Glaucon refers to this knowledge as ‘mortal’. Socrates claims that it is useless unless it is “sought after with a view to be beautiful and good”. All of our knowledge, no matter how expansive, can be futile when used to obtain corruptive power, greed and evil, or when restricted within the borders of obstinacy.

End of Part Two.