Combining Texts

All the ideas for 'The Logic of Boundaryless Concepts', '74: Reply to Colotes' and 'Principles of Arithmetic, by a new method'

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13 ideas

5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
9390 Logic guides thinking, but it isn't a substitute for it [Rumfitt]
     Full Idea: Logic is part of a normative theory of thinking, not a substitute for thinking.
     From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.13)
     A reaction: There is some sort of logicians' dream, going back to Leibniz, of a reasoning engine, which accepts propositions and outputs inferences. I agree with this idea. People who excel at logic are often, it seems to me, modest at philosophy.
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
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
9389 Vague membership of sets is possible if the set is defined by its concept, not its members [Rumfitt]
     Full Idea: Vagueness in respect of membership is consistency with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of a concept.
     From: Ian Rumfitt (The Logic of Boundaryless Concepts [2007], p.5)
     A reaction: I find this view of sets much more appealing than the one that identifies a set with its members. The empty set is less of a problem, as well as non-existents. Logicians prefer the extensional view because it is tidy.
16. Persons / B. Nature of the Self / 7. Self and Body / c. Self as brain controller
5952 Rather than being the whole soul, maybe I am its chief part? [Plutarch]
     Full Idea: Is each of us not the soul, but the chief part of the soul, by which we think and reason and act, all the other parts of soul as well as of body being mere instruments of its power?
     From: Plutarch (74: Reply to Colotes [c.85], §1119)
     A reaction: Socrates is associated with the idea that I am my whole soul (Idea 1650). Plutarch represents an interesting development, which may lead both to the Christian 'soul' and to the Cartesian 'ego'. I think Plutarch is right, but what is the 'soul'?
17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / a. Physicalism critique
5951 If atoms have no qualities, they cannot possibly produce a mind [Plutarch]
     Full Idea: Perception, mind, intelligence and thought cannot so much as be conceived, even with the best will, as arising among void and atoms, things which taken separately have no quality.
     From: Plutarch (74: Reply to Colotes [c.85], §1112)
     A reaction: A nice articulation of the intuition of all anti-physicalists. Plutarch would have to rethink his position carefully if he learned of the sheer number of connections in the brain, and of the theory of natural selection. His challenge remains, though.
20. Action / B. Preliminaries of Action / 1. Intention to Act / c. Reducing intentions
20796 Action needs an affinity for a presentation, and an impulse toward the affinity [Plutarch]
     Full Idea: The sceptics say there are three movements of the soul: presentation, impulse and assent. …And action requires two things: a presentation of something to which one has an affinity, and an impulse toward what is presented as an object of affinity.
     From: Plutarch (74: Reply to Colotes [c.85], 1122c)
     A reaction: Not much reasoning involved in this account, which the sceptics say is compatible with suspension of judgement.
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
5948 The good life involves social participation, loyalty, temperance and honesty [Plutarch]
     Full Idea: To live the good life is to live a life of participation in society, of loyalty to friends, of temperance and honest dealing.
     From: Plutarch (74: Reply to Colotes [c.85], §1108)
     A reaction: 'Participation in society' is the interesting one. This might translate as 'doing your duty', or as 'leading a well-rounded life'. Solitude is wrong if you are indebted to others, and it is unhealthy if you are not. Is solitude really immoral, though?
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / g. Atomism
5950 If only atoms exist, how do qualities arise when the atoms come together? [Plutarch]
     Full Idea: If you accept atomism, you must show how bodies without quality have given rise to qualities of every kind by the mere fact of coming together. For example, how has the quality called 'hot' been imposed on the atoms?
     From: Plutarch (74: Reply to Colotes [c.85], §1111)
     A reaction: This argument is still significant in current philosophy of mind. If temperature is 'mean kinetic energy', you are left wondering where the energy came from, and why minds experience the heat. This is the 'Hard Question'.
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
5955 No one will ever find a city that lacks religious practices [Plutarch]
     Full Idea: A city without holy places and gods, without any observance of prayers, oaths, oracles, sacrifices for blessings received or rites to avert evils, no traveller has ever seen or will ever see.
     From: Plutarch (74: Reply to Colotes [c.85], §1125)
     A reaction: The nearest you might get would be Soviet Moscow, but in 1973 I saw a man there jeering at a woman who was kneeling in the street outside a closed church. Plutarch would be stunned at the decline in religious practices in modern Europe.