10 ideas
15873 | Laws of nature are just records of regularities and correlations, with concepts to make recording them easier [Mach, by Harré] |
Full Idea: For Mach, the laws of nature are simply the compendious record of sensory regularities, correlations of elements. Any additional concepts are no more than symbols or devices for the convenient recording of general sensory patterns. | |
From: report of Ernst Mach (Popular Scientific Lectures [1894], pp.201-5) by Rom Harré - Laws of Nature 2 | |
A reaction: Mach is the high priest of scientific positivism, which is really just hard-line empiricism. |
10528 | Definitions concern how we should speak, not how things are [Fine,K] |
Full Idea: Our concern in giving a definition is not to say how things are by to say how we wish to speak | |
From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310) | |
A reaction: This sounds like an acceptable piece of wisdom which arises out of analytical and linguistic philosophy. It puts a damper on the Socratic dream of using definition of reveal the nature of reality. |
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 |
10529 | If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K] |
Full Idea: Neo-Fregeans have thought that Hume's Principle, and the like, might be definitive of number and therefore not subject to the usual epistemological worries over its truth. | |
From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310) | |
A reaction: This seems to be the underlying dream of logicism - that arithmetic is actually brought into existence by definitions, rather than by truths derived from elsewhere. But we must be able to count physical objects, as well as just counting numbers. |
10530 | Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K] |
Full Idea: The fundamental difficulty facing the neo-Fregean is to either adopt the predicative reading of Hume's Principle, defining numbers, but inadequate, or the impredicative reading, which is adequate, but not really a definition. | |
From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.312) | |
A reaction: I'm not sure I understand this, but the general drift is the difficulty of building a system which has been brought into existence just by definition. |
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 |
10527 | An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K] |
Full Idea: If an abstraction principle is going to be acceptable, then it should not 'inflate', i.e. it should not result in there being more abstracts than there are objects. By this mark Hume's Principle will be acceptable, but Frege's Law V will not. | |
From: Kit Fine (Precis of 'Limits of Abstraction' [2005], p.307) | |
A reaction: I take this to be motivated by my own intuition that abstract concepts had better be rooted in the world, or they are not worth the paper they are written on. The underlying idea this sort of abstraction is that it is 'shared' between objects. |