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.

Definitions

• Meno: is a Socratic dialogue written by Plato. Written in the Socratic dialectic style, it attempts to determine the definition of virtue, or arete, meaning in this case virtue in general, rather than particular virtues (e.g. justice, temperance, etc.). The goal is a common definition that applies equally to all particular virtues. Socrates moves the discussion past the philosophical confusion, or aporia, created by Meno’s paradox with the introduction of new Platonic ideas: the theory of knowledge as recollection, anamnesis, and in the final lines a movement towards Platonic idealism.
• ontology: a branch of philosophy focusing upon the origins, essence and meaning of being

Notes

• Plato was motivated by a gap between the ideas we can conceive and the physical world around us. Eg. perfect justice, beauty, pious, virtuous, etc…everything in the material word has flaws.
• We have some understanding of the perfect ideals, and yet we never find them. Why is this?
• Plato’s answer is that there is a realm of Forms, which contains perfect items like Beauty, Justice, and Piety. The Forms are eternal and unchanging.
• Plato calls the physical realm the world of Becoming, because physical objects are subject to change (for better or for worse) and corruption.
• We understand the physical world or also known as the realm of ‘sights and sounds’ through the senses, but we grasp the Forms only through mental reflection (thinking) because the Forms are objects of thought and invisible.
• Plato uses the experiment (where Socrates lead a slave to the theorem that the square on the diagonal of a given square is double the area of the original square) to support a doctrine that when it comes to geometry-or the world of Being generally-what is called ‘learning’ is actually remembering from a past life, presumably a time when the soul had direct access to the world of Being.
• Plato did hold that the soul is in a third ontological category, with the ability to apprehend both the world of Being and the world of Becoming.
• For Plato, mathematics is a key step in this process. It elevates the soul, reaching beyond the material world to the eternal world of Being.

Definitions

• rationalism: platonism; branch of philosophy which emphasizes reason or intellect, rather than observation or sensory perception, as the basis for knowledge and truth.
• secular: of or relating to the doctrine that rejects religion and religious considerations

Notes

• The problems that occupied mathematicians for centuries, culminating more than 2,000 years later with the result that there are no solutions-the tasks to obtain exact solutions are impossible.
• Thomas Kuhn’s influential Structure of Scientific Revolutions (1970) speaks of revolutions and ‘paradigm shifts’ that make it difficult to understand scientific works of the past. I.e., to understand previous work we have to unlearn our current science and try to immerse ourselves in the overturned world-view.
• However, there’s an exception of this when mathematics is concerned. A contemporary mathematicians does not have to do much (if any) conceptual retooling in order to read and admire Euclid’s Elements.
• Plato stands at the head of a long tradition in philosophy sometimes called rationalism or ‘Platonism’