Combining Texts

All the ideas for 'Through the Looking Glass', 'Letters to Regius' and 'Principles of Arithmetic, by a new method'

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8 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 / A. Nature of Existence / 3. Being / e. Being and nothing

21982	I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L]
	Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people."
	From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7
	A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork.

9. Objects / D. Essence of Objects / 15. Against Essentialism

12251	Substantial forms are not understood, and explain nothing [Descartes]
	Full Idea: Clearly no explanation can be given by these substantial forms for any natural action, since their defenders admit that they are occult and that they do not understand them themselves, ...so they explain nothing.
	From: René Descartes (Letters to Regius [1642], 1642.01), quoted by David S. Oderberg - Real Essentialism 267 n5
	A reaction: [Oderberg gives refs for attack by Locke and Hume, p.66] Descartes' target is Aristotle's hylomorphism. The problem seems to be understanding what Aristotle meant, which is much more than mere 'shape'. More like 'controlling principle'.

29. Religion / B. Monotheistic Religion / 4. Christianity / c. Angels

16772	An angelic mind would not experience pain, even when connected to a human body [Descartes, by Pasnau]
	Full Idea: Descartes points out that an angelic mind, even if causally connected to a human body, would not experience the same sort of bodily sensations; it would, instead, simply observe flesh being torn, like a piece of paper.
	From: report of René Descartes (Letters to Regius [1642], III:493) by Robert Pasnau - Metaphysical Themes 1274-1671 25.6
	A reaction: Does that mean that the angel could not have the experience even if it wanted to have it. So they can't pick up a cup either? So they can't make themselves known to us, even if they are desperate to? So the Annunciation never happened?