12 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) |
| 14759 | A thing is simply a long event, linked by qualities, and spatio-temporal unity [Broad] |
| Full Idea: A thing is simply a long event, throughout the course of which there is either qualitative similarity or continuous qualitative change, together with a characteristic spatio-temporal unity. | |
| From: C.D. Broad (Scientific Thought [1923], 10 'Duration') | |
| A reaction: At least he is trying to give some sort of principle that links the stages of the event together. |
| 11842 | If short-lived happenings like car crashes are 'events', why not long-lived events like Dover Cliffs? [Broad] |
| Full Idea: We call a lightning flash or a motor accident an event, but refuse to apply this to the cliffs of Dover. ...But quantitative differences (of time) give no good grounds for calling one bit of history an event, and refusing the name to another bit. | |
| From: C.D. Broad (Scientific Thought [1923], p.54), quoted by David Wiggins - Sameness and Substance Renewed 2.3 n13 | |
| A reaction: Wiggins calls this proposal a 'terrible absurdity', but it seems to me to demand attention. There is a case to be made for a 'process' to be the fundamental category of our ontology, with stable physical objects seen in that light. |
| 16002 | The self is a combination of pairs of attributes: freedom/necessity, infinite/finite, temporal/eternal [Kierkegaard] |
| Full Idea: A human being is essentially spirit, but what is spirit? Spirit is to be a self. But what is the Self? In short, it is a synthesis of the infinite and the finite, of the temporal and the eternal, of freedom and necessity. | |
| From: Sřren Kierkegaard (Sickness unto Death [1849], p.59) | |
| A reaction: The dense language of his first paragraph was to poke fun at fashionable Hegelian writing. The book gets very lucid afterwards! [SY] |
| 8160 | The present and past exist, but the future does not [Broad, by Dummett] |
| Full Idea: Not only the present but also the past exist, but the future (so long as it is the future) does not. | |
| From: report of C.D. Broad (Scientific Thought [1923]) by Michael Dummett - Thought and Reality 1 | |
| A reaction: This is quite appealing, and seems right if you believe that every truth has a truthmaker, and that there are no truths about the future. And yet the whole misery of people dying is that they cease to exist. |
| 14609 | We could say present and past exist, but not future, so that each event adds to the total history [Broad] |
| Full Idea: One theory accepts the reality of the present and the past, but holds that the future is simply nothing at all. Nothing has happened to the present by becoming past except that fresh slices of existence have been added to the total history of the world. | |
| From: C.D. Broad (Scientific Thought [1923], II) | |
| A reaction: This is now known as Broad's 'Growing Block' view of time. It is tempting to say that neither past nor future exist, but it seems undeniable that statements about the past can be wholly true, unlike those about the future. |
| 22933 | We imagine the present as a spotlight, moving across events from past to future [Broad] |
| Full Idea: We imagine presentness moving, like the spot of light from a policeman's bulls eye traversing the fronts of houses in a street. What is illuminated is present, what was illuminated is past, and what is not yet illuminated is the future. | |
| From: C.D. Broad (Scientific Thought [1923], II) | |
| A reaction: This is the 'moving spotlight' compromise theory, which retains the B-series eternal sequence of ordered events, but adds the A-series privileged present moment. Le Poidevin says Broad represents time twice over. |