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"Platone, in prima luogo, mirà manifestamente ad una fondazione filosofica della dottrina matematica delle proporzioni e dei rapporti." - Hans Krämer
My talk sought to explicate two features of Plato's Cave Allegory. In order of the attention I gave them, they were:
1. The ascent from a lower reality to higher reality, from a lower kind of knowledge to a higher kind of knowledge.
2. The difference between what can be learned from others and true education which can only be acquired by oneself.
This page reviews some of the examples used for those points.
The ascent continues beyond what I describe to ultimate convergent termini:
- to the Good -what is of greatest worth.
- to the True - what is most clear and illuminating.
- to the Beautiful - what is most splendid.
- to Being - what is most reliably real.
To establish facts, to get the features of things, events, etc. rightly connected is one kind of knowing and can be termed information. Examples of information are: this is a triangle; that is Fred; Plato was born in 427 BC; the answer to problem 7 is 4x + 3y = 12; the Frst Amendment protects freedom of speech. Indeed, much of your schooling -- and all your quizzes -- is about information.
Another kind of knowing is understanding. In understanding one comes to know the features that are connected in information. What is a triangle? What is a person? a philosopher? an equation? What is freedom? What is speech? We understand information in terms of our understanding of the features connected in the information. Instead of saying a person understands the features, we usually say a person understands the concepts. A person who does not understand mathematical or political concepts is unable to understand political or mathematical information.
Try the following geometrical example. The directions and diagrams give information in such a way as to help you understand a geometrical concept. When you "see" that the answer is the right answer (and can explain why) you understand the concept. Notice that all the information -- including the right answer -- will not be enough for some of you to "get it." (This is an experience we all have in mathematics -- elsewhere too, but it is most noticeable in mathematics.) Information can be given to you -- all the relevant information -- still, the understanding must come from you. (The example is taken from Plato's Meno.)
| How could you use the figure of the square WHAT to construct a square double in area, that is, a square whose area is in the ratio 2:1 to the area of WHAT? |  |
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Suppose we double the length of each side of WHAT, that is make the ratio of the sides 2:1. The ratio of the areas is then 4:1. Too much |
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Suppose we increase the side of WHAT by half, that is, make the ratio of the sides 3:2. The resulting ratio of areas is 9:4. Still, too much. (Notice the relation between the ratios of sides and the ratios of the corresponding areas.) |
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Suppose, however, we draw the diagonal TH and construct the square THEN. What then? Indeed, the ratio of the areas is 2:1. (There is a problem here: the ratio of the corresponding sides cannot be expressed with numbers. It is ir-rational.) |
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| Side note: The previous diagram also verifies the Pythagorean Theorem for triangle HAT. The square WHAT is a square on the side; the square THEN is the square on the hypotenuse. |  |
Now that you understand - or even especially if you do not
- notice the following aspects of that which is understood.
It is apprehended by the mind; the diagram is suggestive but obviously
is not itself the insight that provides the understanding of the
problem. Moreover, the diagram can be changed, can be inaccurate,
or even confusing, but that does not affect what is understood,
which does not change, is not relative to a (defective) presentation
or to a personal (subjective) point of view. Our apprehension
and appreciation of an insight may change, but not what is understood
in that insight. Finally, what is understood has the stability
of being itself so to speak; we may be confused and temporary,
but it is not.
These aspects can be summarized as follows: what is understood is...
intelligible - apprehended by the mind.
incorporeal - independent of some physical presentation (we need to hear or see it, but it is not identical with what is seen or heard).
unchanging - unlike opinion which depends on us, that which is understood does not change.
self-identical - stays true to itself (note the phrase) and unlike everything physical, does not metamorphose into something else.
one over/in many - it can be recovered (remembered) many times and applied in many cases, remaining the same.
The above bold-faced words describe a "higher reality." The Cave Allegory describes the ascent to that higher reality. The ascent is by means of taking what is familiar and "looking through" it to its higher reality, its original, its concept.
Protagoras once objected to a geometrical diagram. A line does not intersect another at one point but at several -- no matter how fine we draw the diagram. Nor are the lines perfectly straight. Protagoras is right about the diagram, but wrong to use the diagram as an objection to Geometry. I understand the (defective) sketch in terms of what is understood in that sketch: true lines and points. So too any engineer (and scientists most of the time) understands and uses materials not in terms of the actual samples (of steel, plastic, carbon, weight, etc.) but in terms of what those kinds are in themselves. In effect, the things of this world become understood in terms of their defining concepts and the concepts guide the use of the things. (Think of a doctor's diagnosis: the first thing is to establish by symptoms and tests what kind of medical problem the patient has; then the patient is treated in terms of the relevant medical knowledge (theory). The doctor looks "through" the symptoms to their relevant concept, then turns around and treats the symptoms in terms of their relevant concept.
Or again, you understand yourself in terms of concepts taken from philosophy, religion, psychology, politics, advertising or other familiar sources. Experience provides information which you then understand as a mood, a feeling, a meaning in terms of these concepts. Often these concepts are so familiar that one does not notice a difference between the information and the concepts; one does not notice the activity of understanding. Reflection is the effort to recover and attend to the act of understanding.
"Then, if this is true, our view of these matters must be this, that education is not in reality what some people proclaim it to be in their professions. What they aver is that they can put true knowledge into a soul that does not possess it, as if they were inserting vision into the blind eyes. [...] this organ of knowledge must be turned around from the world of becoming together with the entire soul ... until the soul is able to endure the contemplation of essence and the brightest region of being." (518d)