37 ideas
| 15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
| Full Idea: On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1 | |
| A reaction: This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm. |
| 14480 | Maybe analytic truths do not require truth-makers, as they place no demands on the world [Thomasson] |
| Full Idea: It is a venerable view that analytic claims do not require truth-makers, as they place no demands on the world, but this claim has often been challenged. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.4) | |
| A reaction: She offers two challenges (bottom p.68), but I would have thought that the best response is that the meanings of the words themselves constitute truthmakers - perhaps via the essence of each word, as Fine suggests. |
| 17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
| Full Idea: Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets. |
| 17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
| Full Idea: Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections. |
| 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
| Full Idea: Zermelo-Fraenkel axioms: Existence (at least one set); Extension (same elements, same set); Specification (a condition creates a new set); Pairing (two sets make a set); Unions; Powers (all subsets make a set); Infinity (set of successors); Choice | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 |
| 13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
| Full Idea: Zermelo proposed his listed of assumptions (including the controversial Axiom of Choice) in 1908, in order to secure his controversial proof of Cantor's claim that ' we can always bring any well-defined set into the form of a well-ordered set'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1 | |
| A reaction: This is interesting because it sometimes looks as if axiom systems are just a way of tidying things up. Presumably it is essential to get people to accept the axioms in their own right, the 'old-fashioned' approach that they be self-evident. |
| 17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
| Full Idea: I intend to show how the entire theory created by Cantor and Dedekind can be reduced to a few definitions and seven principles, or axioms, which appear to be mutually independent. | |
| From: Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro) | |
| A reaction: The number of axioms crept up to nine or ten in subsequent years. The point of axioms is maximum reduction and independence from one another. He says nothing about self-evidence (though Boolos claimed a degree of that). |
| 13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
| Full Idea: Zermelo's Pairing Axiom superseded (in 1930) his original 1908 Axiom of Elementary Sets. Like Union, its only justification seems to rest on 'limitations of size' and on the 'iterative conception'. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Maddy says of this and Union, that they seem fairly obvious, but that their justification is of prime importance, if we are to understand what the axioms should be. |
| 13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
| Full Idea: Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2 | |
| A reaction: Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity. |
| 13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
| Full Idea: The most characteristic Zermelo axiom is Separation, guided by a new rule of thumb: 'one step back from disaster' - principles of set generation should be as strong as possible short of contradiction. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.4 | |
| A reaction: Why is there an underlying assumption that we must have as many sets as possible? We are then tempted to abolish axioms like Foundation, so that we can have even more sets! |
| 13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
| Full Idea: Zermelo assumes that not every predicate has an extension but rather that given a set we may separate out from it those of its members satisfying the predicate. This is called 'separation' (Aussonderung). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 14471 | Analytical entailments arise from combinations of meanings and inference rules [Thomasson] |
| Full Idea: 'Analytically entail' means entail in virtue of the meanings of the expressions involved and rules of inference. So 'Jones bought a house' analytically entails 'Jones bought a building'. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 01.2) | |
| A reaction: Quine wouldn't like this, but it sounds OK to me. Thomasson uses this as a key tool in her claim that common sense objects must exist. |
| 13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
| Full Idea: In Zermelo's set theory, the Burali-Forti Paradox becomes a proof that there is no set of all ordinals (so 'is an ordinal' has no extension). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by William D. Hart - The Evolution of Logic 3 |
| 18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
| Full Idea: For Zermelo the successor of n is {n} (rather than Von Neumann's successor, which is n U {n}). | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 | |
| A reaction: I could ask some naive questions about the comparison of these two, but I am too shy about revealing my ignorance. |
| 13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
| Full Idea: Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed. |
| 9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
| Full Idea: In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures. | |
| From: report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4 | |
| A reaction: This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another. |
| 14493 | Existence might require playing a role in explanation, or in a causal story, or being composed in some way [Thomasson] |
| Full Idea: A higher standard for saying that entities exist might require that they play an essential role in explanation, or must figure in any complete causal story, or exist according to some uniform and nonarbitrary principle of composition. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 11.2) | |
| A reaction: I am struck by the first of these three. If I am defending the notion that essence depends on Aristotle's account of explanation, then if we add that existence also depends on explanation, we get a criterion for the existence of essences. Yay. |
| 14491 | Rival ontological claims can both be true, if there are analytic relationships between them [Thomasson] |
| Full Idea: Where there are analytic interrelations among our claims, distinct ontological claims may be true without rivalry, redundancy, or reduction. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 10) | |
| A reaction: Thus we might, I suppose, that it is analytically necessary that a lump of clay has a shape, and that a statue be made of something. Interesting. |
| 14489 | Theories do not avoid commitment to entities by avoiding certain terms or concepts [Thomasson] |
| Full Idea: A theory does not avoid commitment to any entities by avoiding use of certain terms or concepts. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.4) | |
| A reaction: This is a salutary warning to those who apply the notion of ontological commitment rather naively. |
| 14485 | Ordinary objects may be not indispensable, but they are nearly unavoidable [Thomasson] |
| Full Idea: I do not argue that ordinary objects are indispensable, but rather that they are (nearly) unavoidable. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09) | |
| A reaction: Disappointing, given the blurb and title of the book, but put in those terms it will be hard to disagree. Clearly ordinary objects figure in the most useful way for us to talk. I wonder whether we have a clear ontology of 'simples' in which they vanish. |
| 14487 | The simple existence conditions for objects are established by our practices, and are met [Thomasson] |
| Full Idea: The existence conditions for ordinary objects are established by our practices, and they are quite minimal, so it is rather obvious that they are fulfilled, and so there are such things. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.3) | |
| A reaction: This is one of her main arguments. The same argument would have worked for witches or ghosts in certain cultures. |
| 21651 | It is analytic that if simples are arranged chair-wise, then there is a chair [Thomasson, by Hofweber] |
| Full Idea: Thomasson argues that the existence of ordinary objects follows analytically from the distribution of simples, assuming that there are any simples. It is an analytic truth that if there are simples arranged chair-wise, then there is a chair. | |
| From: report of Amie L. Thomasson (Ordinary Objects [2007]) by Thomas Hofweber - Ontology and the Ambitions of Metaphysics 07.3 | |
| A reaction: But how do you distinguish when simples are arranged nearly chair-wise from the point where they click into place as actually chair-wise? What is the criterion? |
| 14486 | Eliminativists haven't found existence conditions for chairs, beyond those of the word 'chair' [Thomasson] |
| Full Idea: The eliminativist cannot claim to have 'discovered' some real existence conditions for chairs beyond those entailed by the semantic rules associated with ordinary use of the word 'chair'. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.3) | |
| A reaction: It is difficult to understand atoms arranged 'chairwise' or 'baseballwise' if you don't already know what a chair or a baseball are. |
| 14467 | Ordinary objects are rejected, to avoid contradictions, or for greater economy in thought [Thomasson] |
| Full Idea: Objections to ordinary objects are the Causal Redundancy claim (objects lack causal powers), the Anti-Colocation view (statues and lumps overlap), Sorites arguments, a more economical ontology, or a more scientific ontology. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], Intro) | |
| A reaction: [my summary of two paragraphs] The chief exponents of these views are Van Inwagen and Merricks. Before you glibly accept ordinary objects, you must focus on producing a really strict ontology. These arguments all have real force. |
| 14479 | To individuate people we need conventions, but conventions are made up by people [Thomasson] |
| Full Idea: The conventionalist faces paradox if they hold that conventions are logically prior to people (since this plurality requires conventions of individuation), and people are logically prior to conventions (if they make up the conventions). | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.3) | |
| A reaction: [Sidelle is the spokesman for conventionalism] The best defence would be to deny the second part, and say that conventions emerge from whatever is there, but only conventions can individuate the bits of what is there. |
| 14481 | Wherever an object exists, there are intrinsic properties instantiating every modal profile [Thomasson] |
| Full Idea: In a 'modally plenitudinous' ontology, wherever there is an object at all, there are objects with intrinsic modal properties instantiating every consistent modal profile. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.5) | |
| A reaction: [She cites K.Bennett, Hawley, Rea, Sidelle] I love this. At last a label for the view I have been espousing. I am a Modal Plenitudinist. I must get a badge made. |
| 14482 | If the statue and the lump are two objects, they require separate properties, so we could add their masses [Thomasson] |
| Full Idea: An objection to the idea that statues are not identical to material lumps of stuff is the proliferation of instances of properties shared by those objects. If the mass of the statue is 500kg, and the mass of the lump is 500kg, do we have 1000kg? | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 04.3) | |
| A reaction: [compressed; she cites Rea 1997 and Zimmerman 1995] To wriggle out of this we would have to understand 'object' rather differently, so that an independent mass is not intrinsic to it. I leave this as an exercise for the reader. |
| 14483 | Given the similarity of statue and lump, what could possibly ground their modal properties? [Thomasson] |
| Full Idea: The 'grounding problem' is that given all that the statue and the lump have in common, what could possibly ground their different modal properties? | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 04.4) | |
| A reaction: Their modal properties are, of course, different, because only one of them could survive squashing. Thomasson suggests their difference of sort, but I'm not sure what that means, separately from what they actually are. |
| 14476 | Identity claims between objects are only well-formed if the categories are specified [Thomasson] |
| Full Idea: Identity claims are only well-formed and truth-evaluable if the terms flanking the statement are associated with a certain category of entity each is to refer to, which disambiguates the reference and identity-criteria. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03) | |
| A reaction: The first of her two criteria for identity. She is buying the full Wiggins package. |
| 14477 | Identical entities must be of the same category, and meet the criteria for the category [Thomasson] |
| Full Idea: Identity claims are only true if the entities referred to are of the same category, and meet the criteria of identity appropriate for things of that category. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03) | |
| A reaction: This may be a little too optimistic about having a set of clear-cut and reasonably objective categories to work with, but attempts at establishing metaphysical categories have not gone especially well. |
| 14478 | Modal Conventionalism says modality is analytic, not intrinsic to the world, and linguistic [Thomasson] |
| Full Idea: Modal Conventionalism has at least three theses: 1) modal truths are either analytic truths, or combine analytic and empirical truths, 2) modal properties are not intrinsic features of the world, 3) modal propositions depend on linguistic conventions. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 03.2) | |
| A reaction: [She cites Alan Sidelle 1989 for this view] I disagree mainly with number 2), since I take dispositions to be key intrinsic features of nature, and I interpret dispositions as modal properties. |
| 14466 | A chief task of philosophy is making reflective sense of our common sense worldview [Thomasson] |
| Full Idea: Showing how, reflectively, we can make sense of our unreflective common sense worldview is arguably one of the chief tasks of philosophy. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], Intro) | |
| A reaction: Maybe. The obvious problem is that when you look at weird and remote cultures like the Aztecs, what counts as 'common sense' might be a bit different. She is talking of ordinary objects, though, where her point is reasonable. |
| 14475 | How can causal theories of reference handle nonexistence claims? [Thomasson] |
| Full Idea: Pure causal theories of reference have problems in handling nonexistence claims | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 02.3) | |
| A reaction: This is a very sound reason for shifting from a direct causal baptism view to one in which the baptism takes place by a social consensus. So there is a consensus about 'unicorns', but obviously no baptism. See Evans's 'Madagascar' example. |
| 14474 | Pure causal theories of reference have the 'qua problem', of what sort of things is being referred to [Thomasson] |
| Full Idea: Pure causal theories of reference face the 'qua problem' - that it may be radically indeterminate what the term refers to unless there is some very basic concept of what sort of thing is being referred to. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 02.3) | |
| A reaction: She cites Dummett and Wiggins on this. There is an obvious problem that when I say 'look at that!' there are all sorts of conventions at work if my reference is to succeed. |
| 14488 | Analyticity is revealed through redundancy, as in 'He bought a house and a building' [Thomasson] |
| Full Idea: The analytic interrelations among elements of language become evident through redundancy. It is redundant to utter 'He bought a house and a building', since buying a house analytically entails that he bought a building. | |
| From: Amie L. Thomasson (Ordinary Objects [2007], 09.4) | |
| A reaction: This appears to concern necessary class membership. It is only linguistically redundant if the class membership is obvious. Houses are familiar, uranium samples are not. |
| 541 | Virtue comes more from habit than character [Critias] |
| Full Idea: More men are good through habit than through character. | |
| From: Critias (fragments/reports [c.440 BCE], B09), quoted by John Stobaeus - Anthology 3.29.41 |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |
| Full Idea: When the laws forbade men to commit open crimes of violence, and they began to do them in secret, a wise and clever man invented fear of the gods for mortals, to frighten the wicked, even if they sin in secret. | |
| From: Critias (fragments/reports [c.440 BCE], B25), quoted by Sextus Empiricus - Against the Professors (six books) 9.54 |