23 ideas
21918 | Sufficient Reason can't be proved, because all proof presupposes it [Schopenhauer, by Lewis,PB] |
Full Idea: Schopenhauer said the principle of sufficient reason is not susceptible to proof for the simple reason that it is presupposed in any argument or proof. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §14 p.32-3) by Peter B. Lewis - Schopenhauer 3 | |
A reaction: I would have thought it might be disproved by a counterexample, such as the Gödel sentence of his incompleteness proof, or quantum effects which seem to elude causation. Personally I believe the principle, which I see as the first axiom of philosophy. |
10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1) | |
A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers. |
10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3) |
13044 | Infinity: There is at least one limit level [Potter] |
Full Idea: Axiom of Infinity: There is at least one limit level. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9) | |
A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity. |
10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2) | |
A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?' |
13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5) | |
A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous. |
10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled. |
10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2) | |
A reaction: He cites Gödel's First Incompleteness theorem for this. |
18946 | Unreflectively, we all assume there are nonexistents, and we can refer to them [Reimer] |
Full Idea: As speakers of the language, we unreflectively assume that there are nonexistents, and that reference to them is possible. | |
From: Marga Reimer (The Problem of Empty Names [2001], p.499), quoted by Sarah Sawyer - Empty Names 4 | |
A reaction: Sarah Swoyer quotes this as a good solution to the problem of empty names, and I like it. It introduces a two-tier picture of our understanding of the world, as 'unreflective' and 'reflective', but that seems good. We accept numbers 'unreflectively'. |
10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1) | |
A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth). |
10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8) |
17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms). | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2) | |
A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying. |
21920 | No need for a priori categories, since sufficient reason shows the interrelations [Schopenhauer, by Lewis,PB] |
Full Idea: Schopenhauer dispenses with Kant's a priori categories, since all interrelations between representations are given through the principle of sufficient reason. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
A reaction: I'm not sure how Schopenhauer manages this move. Is it the stoic idea that reality has a logical structure, which can be inferred? Sounds good to me. Further investigation required. |
13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
Full Idea: A set is called a 'relation' if every element of it is an ordered pair. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7) | |
A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related. |
13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here. |
13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go. |
10709 | Priority is a modality, arising from collections and members [Potter] |
Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members. | |
From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background. |
21362 | Necessity is physical, logical, mathematical or moral [Schopenhauer, by Janaway] |
Full Idea: For Schopenauer there are physical necessity, logical necessity, mathematical necessity and moral necessity. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
A reaction: These derive from four modes of explanation, by causes, by grounding in truths or facts, by mathematical reality, and by motives. Not clear why mathematics gets its own necessity. I like metaphysics derived from explanations, though. Necessity makers. |
21361 | For Schopenhauer, material things would not exist without the mind [Schopenhauer, by Janaway] |
Full Idea: Schopenhauer is not a realist about material things, but an idealist: that is, material things would not exist, for him, without the mind. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
A reaction: Janaway places his views as close to Kant's, but it is not clear that Kant would agree that no mind means no world. Did Schopenhauer believe in the noumenon? |
21919 | Object for a subject and representation are the same thing [Schopenhauer] |
Full Idea: To be object for a subject and to be representation is to be one and the same thing. All representations are objects for a subject, all objects for a subject are representations. | |
From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §16 p.41-2), quoted by Peter B. Lewis - Schopenhauer 3 | |
A reaction: This is pure idealism in early Schopenhauer, derived from Kant. Are being 'an object for a subject' and being an object 'in itself' two different things? Compare Idea 21914, written later. I think Nietzsche's 'perspective' representations helps here. |
21917 | The four explanations: objects by causes, concepts by ground, maths by spacetime, ethics by motive [Schopenhauer, by Lewis,PB] |
Full Idea: There are four forms of explanation, depending on their topic. Causes explain objects. Grounding explains concepts, Points and moments explain mathematics. Motives explain ethics. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], §43 p.214) by Peter B. Lewis - Schopenhauer 3 | |
A reaction: [My compression of Lewis's summary. I'm particularly pleased with this. I have done Schopenhauer a huge favour, should anyone ever visit this website]. The quirky account of mathematics derives from Kant. I greatly admire this whole idea. |
21921 | Concepts are abstracted from perceptions [Schopenhauer, by Lewis,PB] |
Full Idea: For Schopenhauer concepts are abstractions from perception, what he calls 'representations of representations', and are linked to the creation of language. | |
From: report of Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813]) by Peter B. Lewis - Schopenhauer 3 | |
A reaction: This is a traditional view which dates back to Aristotle, and which I personally think is entirely correct. These days I am in minority on that. This idea means that (contrary to Kant) perception is not conceptual. |
21363 | Motivation is causality seen from within [Schopenhauer] |
Full Idea: Motivation is causality seen from within. | |
From: Arthur Schopenhauer (Fourfold Root of Princ of Sufficient Reason [1813], p.214), quoted by Christopher Janaway - Schopenhauer 2 'Fourfold' | |
A reaction: This is more illuminating about causation than about motivation, since we can be motivated without actually doing anything. |