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) |
| 5062 | First: there must be reasons; Second: why anything at all?; Third: why this? [Leibniz] |
| Full Idea: We rise to metaphysics by saying 'nothing takes place without a reason', then asking 'why is there something rather than nothing?, and then 'why do things exist as they do?' | |
| From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §7) | |
| A reaction: Wonderful. This is what we pay philosophers for - to attempt to go to the heart of the mystery, and then start formulating the appropriate questions. The question of 'why this?' is the sweetest question. The first one seems a little intractable. |
| 19377 | A monad and its body are living, so life is everywhere, and comes in infinite degrees [Leibniz] |
| Full Idea: Each monad, together with a particular body, makes up a living substance. Thus, there is not only life everywhere, joined to limbs or organs, but there are also infinite degrees of life in the monads, some dominating more or less over others. | |
| From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], 4) | |
| A reaction: Two key ideas: that each monad is linked to a body (which is presumably passive), and the infinite degrees of life in monads. Thus rocks consist of monads, but at an exceedingly low degree of life. They are stubborn and responsive. |
| 19353 | 'Perception' is basic internal representation, and 'apperception' is reflective knowledge of perception [Leibniz] |
| Full Idea: We distinguish between 'perception', the internal state of the monad representing external things, and 'apperception', which is consciousness, or the reflective knowledge of this internal state, not given to all souls, nor at all times to a given soul. | |
| From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §4) | |
| A reaction: The word 'apperception' is standard in Kant. I find it surprising that modern analytic philosophers don't seem to use it when they write about perception. It strikes me as useful, but maybe specialists have a reason for avoiding it. |
| 5061 | Animals are semi-rational because they connect facts, but they don't see causes [Leibniz] |
| Full Idea: There is a connexion between the perceptions of animals, which bears some resemblance to reason: but it is based only on the memory of facts or effects, and not at all on the knowledge of causes. | |
| From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §5) | |
| A reaction: This amounts to the view that animals can do Humean induction (where you see regularities), but not Leibnizian induction (where you see necessities). I say all minds perceive patterns, but only humans can think about the patterns they have perceived. |
| 5063 | Music charms, although its beauty is the harmony of numbers [Leibniz] |
| Full Idea: Music charms us although its beauty only consists in the harmony of numbers. | |
| From: Gottfried Leibniz (Principles of Nature and Grace based on Reason [1714], §17) | |
| A reaction: 'Only'! This is a super-pythagorean view of music, as you might expect from a great mathematician. Did he understand the horrible compromises that had just been made to achieve even-tempered tuning? Patterns are the key, as always. |
| 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. |