13 ideas
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 8978 | Events are made of other things, and are not fundamental to ontology [Bennett] |
| Full Idea: Events are not basic items in the universe; they should not be included in any fundamental ontology...all the truths about them are entailed by and explained and made true by truths that do not involve the event concept. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.12), quoted by Peter Simons - Events 3.1 | |
| A reaction: Given the variable time spans of events, their ability to coincide, their ability to contain no motion, their blatantly conventional component, and their recalcitrance to individuation, I say Bennett is right. |
| 10364 | Facts are about the world, not in it, so they can't cause anything [Bennett] |
| Full Idea: Facts are not the sort of item that can cause anything. A fact is a true proposition (they say); it is not something in the world but is rather something about the world. | |
| From: Jonathan Bennett (Events and Their Names [1988], p.22), quoted by Jonathan Schaffer - The Metaphysics of Causation 1.1 | |
| A reaction: Compare 10361. Good argument, but maybe 'fact' is ambiguous. See Idea 10365. Events are said to be more concrete, and so can do the job, but their individuation also seems to depend on a description (as Davidson has pointed out). |
| 21275 | Unlike a stone, the parts of a watch are obviously assembled in order to show the time [Paley] |
| Full Idea: When we come to inspect a watch we perceive (what we could not discover in a stone) that its several parts are put together for a purpose, to produce motion, and that motion so regulated as to point out the hour of the day. | |
| From: William Paley (Natural Theology [1802], Ch 1) | |
| A reaction: Microscopic examination of the stone would have surprised Paley. Should we infer a geometer because the sun is spherical? Crytals look designed, but are explained by deeper chemistry. |
| 21276 | From the obvious purpose and structure of a watch we must infer that it was designed [Paley] |
| Full Idea: The inference is inevitable that the watch had a maker; that there must have existed, at some time, an artificer or artificers who formed it for the purpose which we find it actually to answer, who designed its use. | |
| From: William Paley (Natural Theology [1802], Ch 1) | |
| A reaction: It rather begs the question to refer to an ordered structure as a 'design'. Why do we think it is absurd to think the the 'purpose' of the sun is to benefit mankind? Suppose we found a freakish natural sundial in the woods. |
| 21277 | Even an imperfect machine can exhibit obvious design [Paley] |
| Full Idea: It is not necessary that a machine be perfect, in order to show with what design it was made. | |
| From: William Paley (Natural Theology [1802], Ch 1) | |
| A reaction: This encounters Hume's point that you will then have to infer that the designer contains similar imperfections. If you look at plagues, famines and mothers dying in childbirth (see Mill), you might wish the designer had never started. |
| 21278 | All the signs of design found in a watch are also found in nature [Paley] |
| Full Idea: Every indication of contrivance, every manifestation of design, which existed in the watch, exists in the works of nature. | |
| From: William Paley (Natural Theology [1802], Ch.3) | |
| A reaction: This is far from obvious. It was crucial to the watch analogy that we immediately see its one self-evident purpose. No one looks at nature and says 'Aha, I know what this is all for'. |
| 21357 | No organ shows purpose more obviously than the eyelid [Paley] |
| Full Idea: The eyelid defends the eye; it wipes it; it closes it in sleep. Are there, in any work of art whatever, purposes more evident than those which this organ fulfils? | |
| From: William Paley (Natural Theology [1802], p.24), quoted by Armand Marie LeRoi - The Lagoon: how Aristotle invented science 031 | |
| A reaction: Nice to have another example, in addition to the watch. He is not wholly wrong, because it is impossible to give an evolutionary account of the development of the eyelid without referring to some sort of teleological aspect. The eyelid has a function. |