13949
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All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R on Peano]
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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.
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From:
comment on Giuseppe Peano (Principles of Arithmetic, by a new method [1889], 11) by Richard Cartwright - Propositions 11
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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.
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5897
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0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]
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Full Idea:
1) 0 is a number; 2) The successor of any number is a number; 3) No two numbers have the same successor; 4) 0 is not the successor of any number; 5) If P is true of 0, and if P is true of any number n and of its successor, P is true of every number.
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From:
report of Giuseppe Peano (works [1890]) by Antony Flew - Pan Dictionary of Philosophy 'Peano'
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A reaction:
Devised by Dedekind and proposed by Peano, these postulates were intended to avoid references to intuition in specifying the natural numbers. I wonder if they could define 'successor' without reference to 'number'.
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20653
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Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson]
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Full Idea:
There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles.
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From:
report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro'
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A reaction:
I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture?
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20992
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Right and wrong concerns what other people cannot reasonably reject [Scanlon]
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Full Idea:
Thinking about right and wrong is, at the most basic level, thinking about what could be justified to others on grounds that they, if appropriately motivated, could not reasonably reject.
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From:
Thomas M. Scanlon (What We Owe to Each Other [1998], Intro)
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A reaction:
The tricky bit is that the acceptance by others must be 'reasonable', so we need a reasonably objective view of rationality. Don't picture your neighbours, picture the locals when you are on holiday in a very different culture. Other Nazis?
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