10 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) |
21846 | Bergson was a rallying point, because he emphasised becomings and multiplicities [Bergson, by Deleuze] |
Full Idea: Bergson was a rallying point for all the opposition, …not so much because of the theme of duration, as of the theory and practice of becoming of all kinds, of coexistent multiplicities. | |
From: report of Henri Bergson (Matter and Memory [1896]) by Gilles Deleuze - A Conversation: what is it? What is it for? I | |
A reaction: The three heroes of Deleuze are Spinoza, Nietzsche and Bergson. All philosophers are either of Being, or of Becoming, I suggest. |
19565 | How could the mind have a link to the necessary character of reality? [Devitt] |
Full Idea: What non-experiential link to reality could support insights into its necessary character? | |
From: Michael Devitt (There is No A Priori (and reply) [2005], 4) | |
A reaction: The key to it, I think, is your theory of mind. If you are a substance dualist, then connecting to such deep things looks fine, but if you are a reductive physicalist then it looks absurdly hopeful. |
19564 | Some knowledge must be empirical; naturalism implies that all knowledge is like that [Devitt] |
Full Idea: It is overwhelmingly plausible that some knowledge is empirical. The attractive thesis of naturalism is that all knowledge is; there is only one way of knowing. | |
From: Michael Devitt (There is No A Priori (and reply) [2005], 1) | |
A reaction: How many ways for us to know seems to depend on what faculties we have. We lump our senses together under a single heading. The arrival of data is not the same as the arrival of knowledge. I'm unconvinced that naturalists like me must accept this. |
21854 | Bergson showed that memory is not after the event, but coexists with it [Bergson, by Deleuze] |
Full Idea: Bergson has shown that memory is not an actual image which forms after the object has been perceived, but a virtual image coexisting with the actual perception of the object. | |
From: report of Henri Bergson (Matter and Memory [1896]) by Gilles Deleuze - The Actual and the Virtual p.114 | |
A reaction: It strikes me as plausible to say that all conscious life is memory. Perceiving the present instant is only possible because it endures for a tiny moment. |