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) |
| 5044 | Reality must be made of basic unities, which will be animated, substantial points [Leibniz] |
| Full Idea: A multiplicity can only be made up of true unities, ..so I had recourse to the idea of a real and animated point, or an atom of substance which must embrace some element of form or of activity in order to make a complete being. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.116) | |
| A reaction: This seems to be a combination of logical atomism and panpsychism. It has a certain charm, but looks like another example of these rationalist speculators overreaching themselves. |
| 16639 | Only individual bodies exist [Bacon] |
| Full Idea: Nothing truly exists in nature beyond individual bodies. | |
| From: Francis Bacon (The New Organon [1620]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 182 | |
| A reaction: [Unusually, Pasnau gives no reference in the text; possibly II:1-2] What this leaves out, from even an auster nominalist ontology, is undifferentiated stuff like water. Even electrons don't seem quite distinct from one another. |
| 16033 | There are only individual bodies containing law-based powers, and the Forms are these laws [Bacon] |
| Full Idea: Though nothing exists in nature except individual bodies which exhibit pure individual acts [powers] in accordance with law…It is this law and its clauses which we understand by the term Forms. | |
| From: Francis Bacon (The New Organon [1620], p.103), quoted by Jan-Erik Jones - Real Essence §3 | |
| A reaction: This isn't far off what Aristotle had in mind, when he talks of forms as being 'principles', though there is more emphasis on mechanisms in the original idea. Note that Bacon takes laws so literally that he refers to their 'clauses'. |
| 21950 | Science must clear away the idols of the mind if they are ever going to find the truth [Bacon] |
| Full Idea: We must clear away the idols and false notions which are now in possession of the human understanding, and have taken deep root therein, and so beset men's minds that truth can hardly find an entrance. | |
| From: Francis Bacon (The New Organon [1620], 38), quoted by Mark Wrathall - Heidegger: how to read 2 | |
| A reaction: [He goes on to list the types of idol] |
| 5045 | No machine or mere organised matter could have a unified self [Leibniz] |
| Full Idea: By means of the soul or form, there is a true unity which is called the 'I' in us; a thing which could not occur in artificial machines, nor in the simple mass of matter, however organised it may be. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.120) | |
| A reaction: I think the unity of consciousness and the unified Self are different phenomena. A wonderful remark about artificial intelligence for 1696! Note the idea of functionalism contained in 'organised'. Personally I see the brain as a 'mass of matter'. |
| 5046 | The soul does know bodies, although they do not influence one another [Leibniz] |
| Full Idea: I do not admit that the soul does not know bodies, although this knowledge arises without their influencing one another. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], Reply 11) | |
| A reaction: He couldn't very well admit this without moving into pure idealism. Presumably it is like "I know her - she'll be in Harrods this morning". I wonder if Satan could steal my body, but my mind continue to believe it was still there? |
| 5043 | To regard animals as mere machines may be possible, but seems improbable [Leibniz] |
| Full Idea: It seems to me that the opinion of those who transform or degrade the lower animals into mere machines, although it seems possible, is improbable, and even against the order of things. | |
| From: Gottfried Leibniz (New System and Explanation of New System [1696], p.116) | |
| A reaction: His target is Descartes. 'Against the order of things' seems to beg the question. What IS the order of things? Only a thorough-going dualist would worry about this question, and that isn't me. |