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) |
| 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. |
| 13165 | Geometrical proofs do not show causes, as when we prove a triangle contains two right angles [Proclus] |
| Full Idea: Geometry does not ask 'why?' ..When from the exterior angle equalling two opposite interior angles it is shown that the interior angles make two right angles, this is not a causal demonstration. With no exterior angle they still equal two right angles. | |
| From: Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics §5 | |
| A reaction: A very nice example. It is hard to imagine how one might demonstrate the cause of the angles making two right angles. If you walk, turn left x°, then turn left y°, then turn left z°, and x+y+z=180°, you end up going in the original direction. |
| 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? |
| 9569 | The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus] |
| Full Idea: It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason. | |
| From: Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55 | |
| A reaction: The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths. |
| 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. |