10 ideas
| 17990 | Instances of minimal truth miss out propositions inexpressible in current English [Hofweber] |
| Full Idea: A standard objection to minimalist truth is the 'incompleteness objection'. Since there are propositions inexpressible in present English the concept of truth isn't captured by all the instances of the Tarski biconditional. | |
| From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 5.3) | |
| A reaction: Sounds like a good objection. |
| 17988 | Quantification can't all be substitutional; some reference is obviously to objects [Hofweber] |
| Full Idea: The view that all quantification is substitutional is not very plausible in general. Some uses of quantifiers clearly seem to have the function to make a claim about a domain of objects out there, no matter how they relate to the terms in our language. | |
| From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 2.1) | |
| A reaction: Robust realists like myself are hardly going to say that quantification is just an internal language game. |
| 17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
| Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro) | |
| A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut']. |
| 10572 | A cut between rational numbers creates and defines an irrational number [Dedekind] |
| Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4) | |
| A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573. |
| 17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
| Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1) | |
| A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting? |
| 18087 | If x changes by less and less, it must approach a limit [Dedekind] |
| Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7 | |
| A reaction: [Kitcher says he 'showed' this, rather than just stating it] |
| 17989 | Since properties have properties, there can be a typed or a type-free theory of them [Hofweber] |
| Full Idea: Since properties themselves can have properties there is a well-known division in the theory of properties between those who take a typed and those who take a type-free approach. | |
| From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 2.2) | |
| A reaction: A typed approach would imply restrictions on what it can be a property of. 'Green' is a property of surfaces, 'dark' is a property of colours. My first reaction is to opt for type-free. |
| 17991 | Holism says language can't be translated; the expressibility hypothesis says everything can [Hofweber] |
| Full Idea: Holism says that nothing that can be said in one language can be said in another one. The expressibility hypothesis says that everything that can be said in one language can be said in every other one. | |
| From: Thomas Hofweber (Inexpressible Properties and Propositions [2006], 6.4) | |
| A reaction: Obviously expressibility would only refer to reasonably comprehensive languages (with basic logical connectives, for example). Personally I vote for the expressibility hypothesis, which Hofweber seems to favour. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |