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) |
| 3916 | Hopi consistently prefers verbs and events to nouns and things [Whorf] |
| Full Idea: Hopi, with its preference for verbs, as contrasted to our own liking for nouns, perpetually turns our propositions about things into propositions about events. | |
| From: Benjamin Lee Whorf (An American Indian model of the Universe [1936], p.63) | |
| A reaction: This should provoke careful thought about ontology - without concluding that it is entirely relative to language. |
| 3917 | Scientific thought is essentially a specialised part of Indo-European languages [Whorf] |
| Full Idea: What we call "scientific thought" is a specialisation of the western Indo-European type of language. | |
| From: Benjamin Lee Whorf (An American Indian model of the Universe [1936], p.246) | |
| A reaction: This is the beginnings of an absurd extreme relativist view of science, based on a confusion about meaning and thought. |
| 3915 | The Hopi have no concept of time as something flowing from past to future [Whorf] |
| Full Idea: A Hopi has no general notion or intuition of time as a smooth flowing continuum in which everything in the universe proceeds at an equal rate, out of a future, through a present, into a past. | |
| From: Benjamin Lee Whorf (An American Indian model of the Universe [1936], p.57) | |
| A reaction: If true, this would not so much support relativism of language as the view that that conception of time is actually false. |
| 7399 | Even without religion, there are many guides to morality [Bacon] |
| Full Idea: Atheism leaves a man to sense, to philosophy, to natural piety, to laws, to reputation; all which may be guides to an outward moral virtue, though religion were not. | |
| From: Francis Bacon (17: Of Superstition [1625], p.52) | |
| A reaction: One might add to Bacon's list 'contracts', or 'rational consistency', or 'self-evident human excellence', or 'natural sympathy'. This is a striking idea, which clearly made churchmen uneasy when atheism began to spread. |