Combining Texts

All the ideas for 'Logic (Port-Royal Art of Thinking)', 'Abstract Objects:intro to Axiomatic Metaphysics' and 'What is Mathematical Truth?'

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9 ideas

7. Existence / C. Structure of Existence / 7. Abstract/Concrete / b. Levels of abstraction

10502	We can rise by degrees through abstraction, with higher levels representing more things [Arnauld,A/Nicole,P]
	Full Idea: I can start with a triangle, and rise by degrees to all straight-lined figures and to extension itself. The lower degree will include the higher degree. Since the higher degree is less determinate, it can represent more things.
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
	A reaction: [compressed] This attempts to explain the generalising ability of abstraction cited in Idea 10501. If you take a complex object and eliminate features one by one, it can only 'represent' more particulars; it could hardly represent fewer.

9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta

10414	Abstract objects are constituted by encoded collections of properties [Zalta, by Swoyer]
	Full Idea: In Zalta's view abstract objects are correlated with collections of properties. ..They encode, as well as exemplify, properties; indeed, an abstract object (such as a Euclidean triangle) is constituted by the properties it encodes.
	From: report of Edward N. Zalta (Abstract Objects:intro to Axiomatic Metaphysics [1983]) by Chris Swoyer - Properties 6.3
	A reaction: If we are going to explain abstract objects with properties, then properties had better not be abstract objects. Zalta has a promising idea if we start from a nominalist and naturalistic view of properties (built from physical powers). 'Encode'?

9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta

10415	Properties make round squares and round triangles distinct, unlike exemplification [Zalta, by Swoyer]
	Full Idea: On Zalta's view, properties with the same encoding extensions are identical, but may be distinct with the same exemplification extension. So the properties of being a round square and a round triangle are distinct, but with the same exemplification.
	From: report of Edward N. Zalta (Abstract Objects:intro to Axiomatic Metaphysics [1983]) by Chris Swoyer - Properties
	A reaction: (For Zalta's view, see Idea 10414) I'm not sure about 'encoding' (cf. Hodes's use of the word), but the idea that an abstract object is just a bunch of possible properties (assuming properties have prior availability) seems promising.

10. Modality / B. Possibility / 1. Possibility

10269	Mathematics eliminates possibility, as being simultaneous actuality in sets [Putnam]
	Full Idea: Mathematics has got rid of possibility by simply assuming that, up to isomorphism anyway, all possibilities are simultaneous actual - actual, that is, in the universe of 'sets'.
	From: Hilary Putnam (What is Mathematical Truth? [1975], p.70), quoted by Stewart Shapiro - Philosophy of Mathematics 7.5

12. Knowledge Sources / B. Perception / 3. Representation

18258	We can only know the exterior world via our ideas [Arnauld,A/Nicole,P]
	Full Idea: We can have knowledge of what is outside us only through the mediation of ideas in us.
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], p.63), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 1 'Conc'

14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence

16784	Forms make things distinct and explain the properties, by pure form, or arrangement of parts [Arnauld,A/Nicole,P]
	Full Idea: The form is what renders a thing such and distinguishes it from others, whether it is a being really distinct from the matter, according to the Schools, or whether it is only the arrangement of the parts. By this form one must explain its properties.
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], III.18 p240), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 27.6
	A reaction: If we ask 'what explains the properties of this thing' it is hard to avoid coming up with something that might be called the 'form'. Note that they allow either substantial or corpuscularian forms. It is hard to disagree with the idea.

15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind

10499	We know by abstraction because we only understand composite things a part at a time [Arnauld,A/Nicole,P]
	Full Idea: The mind cannot perfectly understand things that are even slightly composite unless it considers them a part at a time. ...This is generally called knowing by abstraction. (..the human body, for example).
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
	A reaction: This adds the interesting thought that the mind is forced to abstract, rather than abstraction being a luxury extra feature. Knowledge through analysis is knowledge by abstraction. Also a nice linking of abstraction to epistemology.

15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind

10501	A triangle diagram is about all triangles, if some features are ignored [Arnauld,A/Nicole,P]
	Full Idea: If I draw an equilateral triangle on a piece of paper, ..I shall have an idea of only a single triangle. But if I ignore all the particular circumstances and focus on the three equal lines, I will be able to represent all equilateral triangles.
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
	A reaction: [compressed] They observed that we grasp composites through their parts, and now that we can grasp generalisations through particulars, both achieved by the psychological act of abstraction, thus showing its epistemological power.

15. Nature of Minds / C. Capacities of Minds / 6. Idealisation

10500	No one denies that a line has width, but we can just attend to its length [Arnauld,A/Nicole,P]
	Full Idea: Geometers by no means assume that there are lines without width or surfaces without depth. They only think it is possible to consider the length without paying attention to the width. We can measure the length of a path without its width.
	From: Arnauld / Nicole (Logic (Port-Royal Art of Thinking) [1662], I.5)
	A reaction: A nice example which makes the point indubitable. The modern 'rigorous' account of abstraction that starts with Frege seems to require more than one object, in order to derive abstractions like direction or number. Path widths are not comparatives.