18 ideas
| 1542 | Diogenes of Apollonia was the last natural scientist [Diogenes of Apollonia, by Simplicius] |
| Full Idea: Diogenes of Apollonia was more or less the last of those who made a study of natural science. | |
| From: report of Diogenes (Apoll) (fragments/reports [c.440 BCE], A05) by Simplicius - On Aristotle's 'Physics' 9.25.1 | |
| A reaction: He quotes Theophrastus |
| 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) |
| 18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
| Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking! |
| 18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
| Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses. |
| 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) |
| 489 | Each thing must be in some way unique [Diogenes of Apollonia] |
| Full Idea: No one thing among things subject to change can possibly be exactly like any other thing, without becoming the same thing. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B05), quoted by Simplicius - On Aristotle's 'Physics' 153.8 | |
| A reaction: This is said to be the first ever formulation of the principle of identity of indiscernible. |
| 483 | Start a thesis with something undisputable [Diogenes of Apollonia] |
| Full Idea: In starting any thesis, it seems to me, one should put forward as one's point of departure something incontrovertible. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B01), quoted by Diogenes Laertius - Lives of Eminent Philosophers 09.57 |
| 1544 | Perception must be an internal matter, because we can fail to perceive when we are preoccupied [Diogenes of Apollonia, by Theophrastus] |
| Full Idea: That it is the inner air that perceives, as being a fragment of the god, is shown by the fact that often when our minds are preoccupied with other matters we fail to see or hear. | |
| From: report of Diogenes (Apoll) (fragments/reports [c.440 BCE], A19) by Theophrastus - On the Senses 42 |
| 24042 | The older Diogenes said the soul is air, made of the smallest particles [Diogenes of Apollonia] |
| Full Idea: Diogenes [of Apollonia] took the soul to be air, thnking that of all things air is composed of the smallest particles and is a starting point. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], DK 64), quoted by Aristotle - De Anima 405a21 | |
| A reaction: This suggests that Diogenes of Apollonia was an atomist, if the soul is made of particles. See also Met 984a5, which says Anaxagoras had the same view. |
| 5995 | Diogenes of Apollonia offered the first teleological account of cosmology [Diogenes of Apollonia, by Robinson,TM] |
| Full Idea: Credit for the first clear assertion of teleological explanation in cosmology goes to Diogenes of Apollonia, for whom air is the divine and intelligent ground of the real and disposes things in the best possible way. | |
| From: report of Diogenes (Apoll) (fragments/reports [c.440 BCE]) by T.M. Robinson - Classical Cosmology (frags) | |
| A reaction: The first teleological explanation seems to be based on a conscious mind. There also emerges the possibility of some sort of non-conscious teleology, closer to the laws of physics than to God. |
| 488 | Air is divine, because it is in and around everything, and arranges everything [Diogenes of Apollonia] |
| Full Idea: Air in itself seems to me to be God and to reach everywhere and to arrange everything and to be in everything. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B05), quoted by Simplicius - On Aristotle's 'Physics' 152.22- | |
| A reaction: So water and fire and air have been offered as the ultimate explanans, though no one seems to offer earth, which is too grubby and miserable (and was denied a Form by Plato). 'Air is God' could ground a nice modern religious sect. |
| 484 | Everything is ultimately a variation of one underlying thing [Diogenes of Apollonia] |
| Full Idea: It seems to me that all existing things are created by the alteration of the same thing, and are the same thing. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B02), quoted by Simplicius - On Aristotle's 'Physics' 151.31- |
| 486 | Plants and animals can only come into existence if something fixes their species [Diogenes of Apollonia] |
| Full Idea: No plant could grow out of the earth, and no animal or any other thing could come into being, unless it were so compounded as to be the same. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B02), quoted by Simplicius - On Aristotle's 'Physics' 151.31- |
| 485 | Things must retain their essential nature during change, or mixing would be impossible [Diogenes of Apollonia] |
| Full Idea: If any existing thing were different in its own essential nature, and were not the same thing which was transformed in many ways and changed, in no way could things mix with one another. | |
| From: Diogenes (Apoll) (fragments/reports [c.440 BCE], B02), quoted by Simplicius - On Aristotle's 'Physics' 151.31- |