24 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. |
| 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 |
| 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. |
| 11184 | Aristotelian essentialism is about shared properties, individuating essentialism about distinctive properties [Marcus (Barcan)] |
| Full Idea: An object must have some of its natural properties in this world. Some of those it has in common with objects of some proximate kind (Aristotelian essentialism), and others individuate it from objects of the same kind (individuating essentialism). | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.193) |
| 11181 | Aristotelian essentialism involves a 'natural' or 'causal' interpretation of modal operators [Marcus (Barcan)] |
| Full Idea: Aristotelian essentialism may best be understood on a 'natural' or 'causal' interpretation of the modal operators. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.189) | |
| A reaction: I record this because I very much like the sound of it, though I have yet to fully understand it. |
| 11180 | Essentialist sentences are not theorems of modal logic, and can even be false [Marcus (Barcan)] |
| Full Idea: In the range of modal systems for which Saul Kripke has provided a semantics, no essentialist sentence is a theorem. Furthermore, there are models for which such sentences are demonstrably false. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.188) |
| 11186 | 'Essentially' won't replace 'necessarily' for vacuous properties like snub-nosed or self-identical [Marcus (Barcan)] |
| Full Idea: We would never use 'is essentially' for 'is necessarily' where vacuous properties are concerned, as in 'Socrates is essentially snub-nosed' or 'Socrates is essentially Socrates'. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.193) | |
| A reaction: This simple point does us a huge service in rescuing the word 'essential' from several hundred years of misguided philosophy. |
| 11185 | 'Is essentially' has a different meaning from 'is necessarily', as they often cannot be substituted [Marcus (Barcan)] |
| Full Idea: There seems to be surface synonymy between 'is essentially' and de re occurrences of 'is necessarily', but intersubstitution often fails to preserve sense (as in 'Winston is essentially a cyclist' and 'Winston is necessarily a cyclist'). | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.193) | |
| A reaction: Clearly the two sentences have different meanings, with 'essentially' being a comment about the nature of Winston, and 'necessarily' probably being a comment about the circumstances in which he finds himself. Very nice. See also Idea 11186. |
| 11182 | If essences are objects with only essential properties, they are elusive in possible worlds [Marcus (Barcan)] |
| Full Idea: Some philosophers make a metaphysical shift, by inventing objects (individual concepts, forms, substances) called 'essences', which have only essential properties, and then worry when they can't locate them by rummaging around in possible worlds. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.192) |
| 11183 | The use of possible worlds is to sort properties (not to individuate objects) [Marcus (Barcan)] |
| Full Idea: The usefulness of talk about possible worlds is not for purposes of individuating the object - that can be done in this world; such talk is a way of sorting its properties. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.192) | |
| A reaction: 'Possible worlds are a device for sorting properties' sounds to me like a promising slogan. Ruth Marcus originated rigid designation, before Kripke came up with the label. |
| 11187 | In possible worlds, names are just neutral unvarying pegs for truths and predicates [Marcus (Barcan)] |
| Full Idea: The strategem of talk about possible worlds is that truth assignments of sentences and extensions of predicates may vary, but individual names don't alter their reference (unless they don't refer). They are a neutral peg for descriptions. | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.194) |
| 11189 | Dispositional essences are special, as if an object loses them they cease to exist [Marcus (Barcan)] |
| Full Idea: Being gold or being a man is not accidental. ..Such essences are dispositional properties of a very special kind: if an object had such a property and ceased to have it, it would have ceased to exist or have changed (as if gold is transmuted to lead). | |
| From: Ruth Barcan Marcus (Essential Attribution [1971], p.202) | |
| A reaction: Ruth Marcus is an important founder of modern scientific essentialism, by not only proposing the notion we call rigid designation, but by explicitly defending the essential identities that seem to emerge from modal logic. |
| 14014 | Space alone, and time alone, will fade away, and only their union has an independent reality [Minkowski] |
| Full Idea: Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. | |
| From: Hermann Minkowski (Space and Time [1908], Intro) | |
| A reaction: Notice the qualification that it is a 'kind of' union. Deep confusion arises from exaggerating the analogy between space and time. Craig Bourne remarks (2006:157) that this shows independence of measurement, not of reality |