Combining Texts

All the ideas for 'Events and Their Names', 'Principles of Arithmetic, by a new method' and 'Schopenhauer'

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9 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

13949	All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
	Full Idea: Peano's axioms are categorical (any two models are isomorphic). Some conclude that the concept of natural number is adequately represented by them, but we cannot identify natural numbers with one rather than another of the isomorphic models.
	From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11
	A reaction: This is a striking anticipation of Benacerraf's famous point about different set theory accounts of numbers, where all models seem to work equally well. Cartwright is saying that others have pointed this out.

18113	PA concerns any entities which satisfy the axioms [Peano, by Bostock]
	Full Idea: Peano Arithmetic is about any system of entities that satisfies the Peano axioms.
	From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 6.3) by David Bostock - Philosophy of Mathematics 6.3
	A reaction: This doesn't sound like numbers in the fullest sense, since those should facilitate counting objects. '3' should mean that number of rose petals, and not just a position in a well-ordered series.

17634	Peano axioms not only support arithmetic, but are also fairly obvious [Peano, by Russell]
	Full Idea: Peano's premises are recommended not only by the fact that arithmetic follows from them, but also by their inherent obviousness.
	From: report of Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

15653	We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach on Peano]
	Full Idea: Peano Arithmetic cannot derive its own consistency from within itself. But it can be strengthened by adding this consistency statement or by stronger axioms (particularly ones partially expressing soundness). These are known as Reflexion Principles.
	From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 1.2) by Volker Halbach - Axiomatic Theories of Truth (2005 ver) 1.2

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism

17635	Arithmetic can have even simpler logical premises than the Peano Axioms [Russell on Peano]
	Full Idea: Peano's premises are not the ultimate logical premises of arithmetic. Simpler premises and simpler primitive ideas are to be had by carrying our analysis on into symbolic logic.
	From: comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], p.276) by Bertrand Russell - Regressive Method for Premises in Mathematics p.276

7. Existence / B. Change in Existence / 4. Events / c. Reduction of events

8978	Events are made of other things, and are not fundamental to ontology [Bennett]
	Full Idea: Events are not basic items in the universe; they should not be included in any fundamental ontology...all the truths about them are entailed by and explained and made true by truths that do not involve the event concept.
	From: Jonathan Bennett (Events and Their Names [1988], p.12), quoted by Peter Simons - Events 3.1
	A reaction: Given the variable time spans of events, their ability to coincide, their ability to contain no motion, their blatantly conventional component, and their recalcitrance to individuation, I say Bennett is right.

11. Knowledge Aims / C. Knowing Reality / 3. Idealism / d. Absolute idealism

21912	Fichte, Schelling and Hegel rejected transcendental idealism [Lewis,PB]
	Full Idea: Fichte, Schelling and Hegel were united in their opposition to Kant's Transcendental Idealism.
	From: Peter B. Lewis (Schopenhauer [2012], 3)
	A reaction: That is, they preferred genuine idealism, to the mere idealist attitude Kant felt that we are forced to adopt.

21911	Fichte, Hegel and Schelling developed versions of Absolute Idealism [Lewis,PB]
	Full Idea: At the University of Jena, Fichte, Hegel and Schelling critically developed aspects of Kant's philosophy, each in his own way, thereby giving rise to the movement known as Absolute Idealism, see reality as universal God-like self-consciousness.
	From: Peter B. Lewis (Schopenhauer [2012], 2)
	A reaction: Is asking how anyone can possibly have believed such a bizarre and ridiculous idea a) uneducated, b) stupid, c) unimaginative, or d) very sensible? It sounds awfully like Spinoza's concept of God. Also Anaxagoras.

26. Natural Theory / C. Causation / 8. Particular Causation / b. Causal relata

10364	Facts are about the world, not in it, so they can't cause anything [Bennett]
	Full Idea: Facts are not the sort of item that can cause anything. A fact is a true proposition (they say); it is not something in the world but is rather something about the world.
	From: Jonathan Bennett (Events and Their Names [1988], p.22), quoted by Jonathan Schaffer - The Metaphysics of Causation 1.1
	A reaction: Compare 10361. Good argument, but maybe 'fact' is ambiguous. See Idea 10365. Events are said to be more concrete, and so can do the job, but their individuation also seems to depend on a description (as Davidson has pointed out).