11 ideas
| 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 |
| 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 |
| 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 |
| 13168 | My formal unifying atoms are substantial forms, which are forces like appetites [Leibniz] |
| Full Idea: To find real entities I had recourse to a unified formal atom. Hence I rehabilitated the substantial forms in a way to render them intelligible. I found that their nature consists in force, from which follows something analogous to sensation and appetite. | |
| From: Gottfried Leibniz (New system of communication of substances [1695], p.139) | |
| A reaction: [several lines are here compressed] This passage sums up the key to Leibniz's essentialism, which I take to be a connection between Aristotelian form and the physicists' notion of force. This gives us a modern version of Aristotelianism for science. |
| 13169 | I call Aristotle's entelechies 'primitive forces', which originate activity [Leibniz] |
| Full Idea: Forms establish the true general principles of nature. Aristotle calls them 'first entelechies'; I call them, perhaps more intelligibly, 'primitive forces', which contain not only act or the completion of possibility, but also an original activity. | |
| From: Gottfried Leibniz (New system of communication of substances [1695], p.139) | |
| A reaction: As in Idea 13168, I take Leibniz to be unifying Aristotle with modern science, and offering an active view of nature in tune with modern scientific essentialism. Laws arise from primitive force, and are not imposed from without. |
| 13170 | The analysis of things leads to atoms of substance, which found both composition and action [Leibniz] |
| Full Idea: There are only atoms of substance, that is, real unities absolutely destitute of parts, which are the source of actions, the first absolute principles of the composition of things, and, as it were, the final elements in the analysis of substantial things. | |
| From: Gottfried Leibniz (New system of communication of substances [1695], p.142) | |
| A reaction: I like this because it addresses the pure issue of the identity of an individuated object, but also links it with an active view of nature, and not some mere inventory of objects. |
| 13171 | Substance must necessarily involve progress and change [Leibniz] |
| Full Idea: The nature of substance necessarily requires and essentially involves progress or change, without which it would not have the force to act. | |
| From: Gottfried Leibniz (New system of communication of substances [1695], p.144) | |
| A reaction: Bravo. Most metaphysical musings regarding 'substance' seem entirely wrapped up in the problem of pure identity, and forget about the role of objects in activity and change. |
| 7825 | The politics of Leibniz was the reunification of Christianity [Stewart,M] |
| Full Idea: The politics of Leibniz may be summed up in one word: theocracy. The specific agenda motivating much of his work was to reunite the Protestant and Catholic churches | |
| From: Matthew Stewart (The Courtier and the Heretic [2007], Ch. 5) | |
| A reaction: This would be a typical project for a rationalist philosopher, who thinks that good reasoning will gradually converge on the one truth. |
| 13167 | We need the metaphysical notion of force to explain mechanics, and not just extended mass [Leibniz] |
| Full Idea: Considering 'extended mass' alone was not sufficient to explain the principles of mechanics and the laws of nature, but it is necessary to make use of the notion of 'force', which is very intelligible, despite belonging in the domain of metaphysics. | |
| From: Gottfried Leibniz (New system of communication of substances [1695], p.139) | |
| A reaction: We may find it surprising that force is a metaphysical concept, but that is worth pondering. It is a mysterious notion within physics. Notice the emphasis on what explains, and what is intelligible. He sees Descartes's system as too passive. |