Combining Texts

All the ideas for 'fragments/reports', 'What Numbers Are' and 'Understanding'

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6 ideas

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP]
     Full Idea: The Löwenheim-Skolem theorem tells us that any theory with a true interpretation has a model in the natural numbers.
     From: Nicholas P. White (What Numbers Are [1974], V)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Finite cardinalities don't need numbers as objects; numerical quantifiers will do [White,NP]
     Full Idea: Statements involving finite cardinalities can be made without treating numbers as objects at all, simply by using quantification and identity to define numerically definite quantifiers in the manner of Frege.
     From: Nicholas P. White (What Numbers Are [1974], IV)
     A reaction: [He adds Quine 1960:268 as a reference]
11. Knowledge Aims / A. Knowledge / 2. Understanding
Unlike knowledge, you can achieve understanding through luck [Grimm]
     Full Idea: It may be that understanding is compatible with luck, in a way that knowledge is not.
     From: Stephen R. Grimm (Understanding [2011], 3)
     A reaction: [He cites Kvanvig and Prichard] If so, then we cannot say that knowledge is a lesser type of understanding. If you ask a trusted person how a mechanism works, and they have a wild guess that is luckily right, you would then understand it.
'Grasping' a structure seems to be modal, because we must anticipate its behaviour [Grimm]
     Full Idea: 'Graspng' a structure would seem to bring into play something like a modal sense or ability, not just to register how things are, but also to anticipate how certain elements of the system would behave.
     From: Stephen R. Grimm (Understanding [2011], 2)
     A reaction: In the case of the chronology of some historical events, talking of 'grasping' or 'understanding' seems wrong because the facts are static and invariant. That seems to support the present idea. But you might 'understand' a pattern if you can reproduce it.
You may have 'weak' understanding, if by luck you can answer a set of 'why questions' [Grimm]
     Full Idea: There may be a 'weak' sense of understanding, where all you need to do is to be able to answer 'why questions' successfully, where one might have come by this ability in a lucky way.
     From: Stephen R. Grimm (Understanding [2011], 3)
     A reaction: We can see this point (in Idea 19691), but the idea that one could come by true complex understanding of something by purely lucky means is a bit absurd. Surely you would get one or two why questions wrong? 100%, just by luck?
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Musical performance can reveal a range of virtues [Damon of Ath.]
     Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice.
     From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where?