Combining Texts

All the ideas for 'Introduction to Russell's Theory of Types', 'Paradoxes of the Infinite' and 'Introspection'

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6 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

9987	An aggregate in which order does not matter I call a 'set' [Bolzano]
	Full Idea: An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential difference, I call a 'set'.
	From: Bernard Bolzano (Paradoxes of the Infinite [1846], §4), quoted by William W. Tait - Frege versus Cantor and Dedekind IX
	A reaction: The idea of 'sets' was emerging before Cantor formalised it, and clarified it by thinking about infinite sets. Nowadays we also have 'ordered' sets, which rather contradicts Bolzano, and we also expect the cardinality to be determinate.

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility

18170	The Axiom of Reducibility is self-effacing: if true, it isn't needed [Quine]
	Full Idea: The Axiom of Reducibility is self-effacing: if it is true, the ramification it is meant to cope with was pointless to begin with.
	From: Willard Quine (Introduction to Russell's Theory of Types [1967], p.152), quoted by Penelope Maddy - Naturalism in Mathematics I.1
	A reaction: Maddy says the rejection of Reducibility collapsed the ramified theory of types into the simple theory.

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

10856	A truly infinite quantity does not need to be a variable [Bolzano]
	Full Idea: A truly infinite quantity (for example, the length of a straight line, unbounded in either direction) does not by any means need to be a variable.
	From: Bernard Bolzano (Paradoxes of the Infinite [1846]), quoted by Brian Clegg - Infinity: Quest to Think the Unthinkable §10
	A reaction: This is an important idea, followed up by Cantor, which relegated to the sidelines the view of infinity as simply something that could increase without limit. Personally I like the old view, but there is something mathematically stable about infinity.

12. Knowledge Sources / B. Perception / 8. Adverbial Theory

5691	The adverbial account of sensation says not 'see a red image' but be 'appeared to redly' [Shoemaker]
	Full Idea: Some who reject the act-object conception of sensation favour an 'adverbial' account, where (instead of the act of 'seeing a red image') it is better to speak of 'being appeared to redly'.
	From: Sydney Shoemaker (Introspection [1994], p.398)
	A reaction: The point is that you couldn't perceive without a colour (or travel without a speed), so the qualifying adverb is intrinsic to the process, not a separate object. The adverbial theory will imply a fairly minimal account of universals.

16. Persons / C. Self-Awareness / 1. Introspection

5687	For true introspection, must we be aware that we are aware of our mental events? [Shoemaker]
	Full Idea: Some writers distinguish introspection from a pre-introspective awareness of mental phenomena, saying one is not properly introspecting unless one is not only aware of the phenomena, but aware that one is aware of them.
	From: Sydney Shoemaker (Introspection [1994], p.395)
	A reaction: The test question might be what we think animals do. I think I agree with the 'writers'. You are either just aware of the contents or qualia or images of thought, which is not introspection, or you become introspectively aware that you are having them.

5688	Empirical foundationalism says basic knowledge is self-intimating, and incorrigible or infallible [Shoemaker]
	Full Idea: Foundationalist epistemology takes all empirical knowledge to be grounded in the introspective knowledge each mind has of its own states, …holding that introspective judgements are 'incorrigible' or 'infallible', and mental states are 'self-intimating'.
	From: Sydney Shoemaker (Introspection [1994], p.396)
	A reaction: Descartes' foundationalist Cogito also seems to be based on introspection, making introspection the essence of all foundationalism. The standard modern view is that introspective states are incorrigible, but not infallible.