Erudite Relics of Loc Huu Ho

Definitions

• adept: is an individual identified as having attained a specific level of knowledge, skill, or aptitude in doctrines relevant to a particular author or organization
• axiomatic: evident without proof or argument; of or pertaining to an axiom; obvious (layman)
  
• exegetical: related to an exegesis, which is the interpretation and understanding of a text on the basis of the text itself
• gnomon: pronounced NO-mon, a Greek word meaning “the one who knows.” The gnomon is the pointer on a sundial, the part of the sundial that “knows” the time
• interlocutor: a person who takes part in a conversation
• par excellence: being the best of its kind; being a quintessential example of the kind in question
• truth-value: a proposition’s truth-value is its being true or its being false

Notes

• Mathematics, or at least geometry, provides a straightforward instance of the gap between the flawed material world around us and the serene, ideal perfect world of thought.
• Plato believed that the propositions of geometry are objectively true or false, independent of the human mind, language, and so on of mathematicians. He believed that geometrical objects are like Forms and are in the world of Being where it is not physical, and that they are eternal and unchanging. He would thus reject the above suggestion that geometric objects exist in physical space.
• Refer to the end of Book 6 of the Republic Plato gives a metaphor of a divided line: the divisions are unequal, with the Forms getting the largest space. The following double proportion holds: Forms are to mathematical objects as physical objects are to reflections, as Being (i.e. Forms plus mathematical objects) is to Becoming (i.e., physical objects and reflections). Although Plato does not mention this, it follows that the ‘mathematical objects’ segment is exactly the same size as the ‘physical objects’ segment.
• Geometry is not about anything in the physical world, the world of Becoming, and we do not apprehend geometric objects via the senses. With the exception that some physical objects approximate Euclidean figures, but geometric theorems do not apply to these approximations.
• We are in position to better understand Plato’s remark in the passage from Book 7 of the Republic, quoted in chapter 1:

[The] science [of geometry] is in direct contradiction with the language employed by its adepts…Their language is most ludicrous…for they speak as if they were doing something and as if all their words were directed toward action…[They talk] of squaring and applying and adding and the like…whereas in fact the real object of the entire subject is… knowledge…of what eternally exists, not of anything that comes to be this or that at some time and ceases to be. (Plato, 1961, 527a in the standard numbering)

(510d) You…know how [geometers] make use of visible figures and discourse about them, though what they really have in mind is the originals of which these figures are images. They are not reasoning, for instance, about this particular square and diagonal which they have drawn, but about the Square and the Diagonal; and so in all cases. The diagrams they draw and the models they make are actual things, which may have their shadows or images in water; but now they serve in their turn as images, while the student is seeking to behold those realities which only thought can comprehend.

• Most Platonists maintained that geometrical knowledge is a priori, independent of sensory experience. It may be that some sensory experience is necessary to grasp the relevant concepts, or we may need drawn diagrams as a visual aid to the mind, or perhaps to awaken our minds to the eternal and unchanging geometric realm of Euclidean space.
• The details of Plato’s views concerning arithmetic and algebra are not as straightforward as his account of geometry, but the overall picture is the same. We see that arithmetic, like geometry, applies to the material world only approximately, or only to the extent that objects can be distinguished from each other.
• Several ancient sources distinguish the theory of numbers (world of Being), called ‘arithmetic’ from the theory of calculations (world of Becoming), called ‘logistic’.
• It is through the study of the numbers themselves, and the relations among numbers, that the soul is able to grasp the nature of numbers as they are in themselves.