45 ideas
14721 | Metaphysical enquiry can survive if its conclusions are tentative [Sider] |
Full Idea: Metaphysical enquiry can survive if we are willing to live with highly tentative conclusions. | |
From: Theodore Sider (Four Dimensionalism [2001], Intro) | |
A reaction: Nice. Nothing alienates the rather literal scientific sort of mind quicker that bold, dogmatic and even arrogant assertions about metaphysics. But to entirely close down metaphysical speculation for that reason is absurd. |
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. |
14760 | Four-dimensionalism sees things and processes as belonging in the same category [Sider] |
Full Idea: Four-dimensionalism does not respect a deep difference between thing-talk and process-talk, because it tends to place events and things in the same ontological category. | |
From: Theodore Sider (Four Dimensionalism [2001], 6.1) | |
A reaction: He then quotes Broad, Idea 14759. This idea is the best reason yet for being sympathetic to the four-dimensionalist view, because I think processes really must have a central place in any decent ontology. |
14194 | Proper ontology should only use categorical (actual) properties, not hypothetical ones [Sider] |
Full Idea: A proper ontology should invoke only categorical, or occurrent, properties and relations. Categorical properties involve what objects are actually like, whereas hypothetical properties 'point beyond' their instances. | |
From: Theodore Sider (Four Dimensionalism [2001], 2.3) | |
A reaction: This spectacularly leaves out powers and dispositions, which are actual properties which 'point beyond' their instances! This is the nub of the powers debate, and the most interesting topic in modern metaphysics. |
14745 | If sortal terms fix the kind and the persistence conditions, we need to know what kinds there are [Sider] |
Full Idea: Followers of the view that every entity is associated with some sortal term that answers the question 'what kind of thing is this?', and determines its persistence conditions, must answer the question what kinds of entity there are. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.3) | |
A reaction: [He explicitly refers to David Wiggins here] In other words Wiggins has got it the wrong way round, which is my own view of his theory. Sortal terms don't grow on the trees in the Garden of Eden, available for applications. |
14740 | If Tib is all of Tibbles bar her tail, when Tibbles loses her tail, two different things become one [Sider] |
Full Idea: This powerful puzzle (known to the Stoics, introduced by Geach, popularised by Wiggins) has a cat Tibbles and a proper part Tib, which is all of Tibbles except the tail. If Tibbles loses her tail, the two were distinct, but they now coincide. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.1) | |
A reaction: [compressed] Compare a few people leave a football ground, and what was a large part of the crowd becomes the whole of the crowd. Which suggests that there is no problem if cats are like crowds. But we don't like that view of cats. |
14752 | Artists 'create' statues because they are essentially statues, and so lack identity with the lump of clay [Sider] |
Full Idea: Presumably it is claimed that the artist 'created' the statue because the object created is essentially a statue, and thus cannot be identified with the unformed lump of clay with which the artist began. | |
From: Theodore Sider (Four Dimensionalism [2001]) | |
A reaction: This is based on Burke's views. This is sortal essentialism, rather than my own view of essence as an inner explanatory mechanism or form. If an old abstract sculpture was no longer recognised as a statue, would it necessarily still be a statue? |
14743 | The stage view of objects is best for dealing with coincident entities [Sider] |
Full Idea: There are numerous cases in which there is pressure to admit coincident entities. The best way of coming to grips with this, I think, invokes the stage view. ...In the worm theory, coincident objects are no more mysterious than overlapping roads. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.1) | |
A reaction: At this point I get nervous if in order to 'get to grips' with a phenomenon which is hard to articulate but obvious to common sense, we have to invoke a rather startling metaphysics that completely upends the common sense we started with. |
14747 | 'Composition as identity' says that an object just is the objects which compose it [Sider] |
Full Idea: 'Composition as identity' says that when a thing, x, is composed of some other objects, the ys, then this is a kind of identity between the x and the ys. The industrial-strength version says object x just is the ys. Lewis says it is just an analogy. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.3) | |
A reaction: I am averse to such a doctrine, as is Leibniz, with his insistence that an aggregate is not a unity. There has to be some sort of principle that bestows oneness on a many. I take this to be structural, and is an elucidation of hylomorphism. |
14757 | Mereological essentialism says an object's parts are necessary for its existence [Sider] |
Full Idea: Mereological essentialism says that an object's parts are necessary for its existence. ....It is literally never correct to say that an thing survives a change in its parts. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.7) | |
A reaction: Chisholm is well known for proposing this view. Sider adds a possible toughening clause, that the parts are also sufficient for the object's existence. This is a philosophers' notion of identity, not the normal English language concept. |
14727 | Three-dimensionalists assert 'enduring', being wholly present at each moment, and deny 'temporal parts' [Sider] |
Full Idea: Three-dimensionalists say that things have no 'temporal parts', that they 'endure', and that they are wholly present at every moment of their careers. | |
From: Theodore Sider (Four Dimensionalism [2001], 3) | |
A reaction: An obvious problem case for being wholly present would be the building and fitting of a large ship, where it might seem to be present before it was wholly present. |
14738 | Some might say that its inconsistency with time travel is a reason to favour three-dimensionalism [Sider] |
Full Idea: Some might even regard inconsistency with time travel as an advantage of three-dimensionalism, as a vindication of a prior belief that time travel is impossible! I see no merit in these claims. | |
From: Theodore Sider (Four Dimensionalism [2001], 7.2) | |
A reaction: I do! Sider cheerfully says that there are good reasons to believe that time travel is possible, and then use this possibility to support his four-dimensional view, but I personally doubt his assumption. The evidence for time travel is flimsy and obscure. |
14726 | Four-dimensionalists assert 'temporal parts', 'perduring', and being spread out over time [Sider] |
Full Idea: Four-dimensionalists say that things have 'temporal parts', that they 'perdure', and that they are spread out over time. | |
From: Theodore Sider (Four Dimensionalism [2001], 3) |
14728 | 4D says intrinsic change is difference between successive parts [Sider] |
Full Idea: For four-dimensionalists intrinsic change is difference between successive temporal parts. | |
From: Theodore Sider (Four Dimensionalism [2001], 3.2) | |
A reaction: This attempts a reply to the commonest criticism of four-dimensionalism - that you can't explain change if you don't have one enduring thing which undergoes the change. I get stuck of the question 'how big (temporally) is a part?'. |
14729 | 4D says each spatiotemporal object must have a temporal part at every moment at which it exists [Sider] |
Full Idea: Four-dimensionalism may be formulated as the claim that, necessarily, each spatiotemporal object has a temporal part at every moment at which it exists. | |
From: Theodore Sider (Four Dimensionalism [2001], 3.2) | |
A reaction: If there were tiny quantum gaps between temporal parts, that would presumably ruin the story. On this view an object has to be a 'worm', to be the thing which has the parts. |
14730 | Temporal parts exist, but are not prior building blocks for objects [Sider] |
Full Idea: My four-dimensionalism implies the existence of temporal parts, but not that those parts are more fundamental, nor that the object is 'constructed' from its parts, nor that identity over time is reducible to parts. | |
From: Theodore Sider (Four Dimensionalism [2001], 3.2) | |
A reaction: That's a rather negative account of temporal parts, which makes you ask what their positive role could be. Do they contribute anything to our understanding of a temporally extended object? |
14731 | Temporal parts are instantaneous [Sider] |
Full Idea: Unless otherwise noted, I will think of temporal parts as being instantaneous. | |
From: Theodore Sider (Four Dimensionalism [2001], 3.2) | |
A reaction: This comes up against all the Augustinian worries about the intrinsic nature of time. How many temporal parts does a typical object possess? Is a third temporal part always to be found between any two of them? How do they 'connect'? |
14758 | How can an instantaneous stage believe anything, if beliefs take time? [Sider] |
Full Idea: How can an instantaneous stage believe anything? Beliefs take time. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.8) | |
A reaction: Sider's four-dimensionalist answer is that the belief is embodied in the earlier counterparts, making belief a 'highly relational property'. I am not impressed by this answer to the very nice problem which he has raised. It's a problem for 3D, too. |
14762 | Four-dimensionalism says temporal parts are caused (through laws of motion) by previous temporal parts [Sider] |
Full Idea: The sensible four-dimensionalist will claim that current temporal parts are caused to exist by previous temporal parts. The laws that govern this process are none other than the familiar laws of motion. | |
From: Theodore Sider (Four Dimensionalism [2001], 6.3) | |
A reaction: I keep struggling with the instantaneous natural of temporal parts, and now I find that they have to do the job of being causal relata. When do they do their job? They've gone home before they've finished clocking in. Continuance requires motion? |
14741 | The ship undergoes 'asymmetric' fission, where one candidate is seen as stronger [Sider] |
Full Idea: The Ship of Theseus seems to be a case of 'asymmetric' fission (where one resultant entity has a stronger claim). Many see the continuously rebuilt ship as the stronger candidate, but each candidate, without the other, would be the original ship. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.1) |
14754 | If you say Leibniz's Law doesn't apply to 'timebound' properties, you are no longer discussing identity [Sider] |
Full Idea: If someone is in pain at t1 and not at t2, we might restrict Leibniz's Law so as not to apply to 'timebound' properties, ..but this is deeply unsatisfying, ...and forfeits one's claim to be discussing identity. The demands of identity are high. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.5) | |
A reaction: [on Myro 1986] Sider's response is unsatisfying. It means a thing loses its identity (with itself?) if it has even a tiny fluctuating in its properties. Quantum changes then destroy all notions of identity. English-speakers don't use 'identity' like that. |
14763 | Counterparts rest on similarity, so there are many such relations in different contexts [Sider] |
Full Idea: A counterpart relation is a similarity relation. Since there are different dimensions of similarity, there are different counterpart relations. | |
From: Theodore Sider (Four Dimensionalism [2001], 6.4) |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
14725 | Maybe motion is a dynamical quantity intrinsic to a thing at a particular time [Sider] |
Full Idea: There is an alternative to the Russellian 'at-at' theory of motion, according to which dynamical quantities are intrinsic to times. Whether and how an object is moving at a time is a fact about what that object is like then. | |
From: Theodore Sider (Four Dimensionalism [2001], 2.2) | |
A reaction: I think I find this quite appealing, because there is too much of a tendency to think of objects as passive and inert, with laws, forces, motions etc. imposed from the outside. But nature is active and dynamic. However, motion can't be wholly intrinsic. |
14735 | Space is 3D and lacks a direction; time seems connected to causation [Sider] |
Full Idea: Unlike time, space has three dimensions and lacks a distinguishing direction; unlike space, time seems to be specially connected with causation. | |
From: Theodore Sider (Four Dimensionalism [2001], 4.5) | |
A reaction: These strike me as nice reasons to doubt (what I already prima facie doubt) that there is a single manifold that is 'space-time', for all that twentieth century physics tells us it is so. A century is a mere click of a clock where truth is concerned. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |
14722 | Between presentism and eternalism is the 'growing block' view - the past is real, the future is not [Sider] |
Full Idea: Intermediate between the polar opposites of presentism and eternalism is the view (defended by Broad 1923 and Tooley 1997) that the past is real but the future is not. Reality consists of a growing four-dimensional manifold, the 'growing block universe'. | |
From: Theodore Sider (Four Dimensionalism [2001], 2.1) | |
A reaction: The obvious and plausible basis for this is that statements about the past seem to have truthmakers, but statements about the future lack them. Does a truth always require ontological commitment? Death is cessation of existence. |
14724 | Presentists must deny truths about multiple times [Sider] |
Full Idea: The presentist must deny the truth of everyday claims that concern multiple times taken together. | |
From: Theodore Sider (Four Dimensionalism [2001], 2.2) | |
A reaction: This rests on the extent to which every truth has an ontological commitment. You can deny the literal existence of multiple times without denying such truths. |
14756 | For Presentists there must always be a temporal vantage point for any description [Sider] |
Full Idea: The Presentist acknowledges that no atemporal description of the case can be given; a vantage point must be chosen for any description. | |
From: Theodore Sider (Four Dimensionalism [2001], 5.5) | |
A reaction: This is because Presentists are committed to tense, which have to be either explicit or implicit in any sentence. But what of famously 'timeless' truths such as '2 and 2 are 4'? |
14723 | Talk using tenses can be eliminated, by reducing it to indexical connections for an utterance [Sider] |
Full Idea: The temporal reductionist claims that tensed locutions are indexical - 'present' being the time of utterance etc. This generalises to say that nothing corresponding to tense need be admitted as a fundamental feature of the world. | |
From: Theodore Sider (Four Dimensionalism [2001], 2.1) | |
A reaction: [He particular cites Mellor for this view] Highly implausible. I very much doubt whether it is possible to explain the indexicality of a word like 'now' without referring to tenses. Does time only exist when sentences and thoughts occur? |
14736 | The B-theory is adequate, except that it omits to say which time is present [Sider] |
Full Idea: The B-theoretic description of the world is completely adequate except that it leaves out information about which time is present. | |
From: Theodore Sider (Four Dimensionalism [2001], 4.6) | |
A reaction: This strikes me as a pretty basic deficiency. How could there a time which lacked a present moment? The present is when things happen. How would it qualify as time at all if it lacked past, present and future? |
14734 | The B-series involves eternalism, and the reduction of tense [Sider] |
Full Idea: The B-series has two components: eternalism - the thesis that all future entities are real - and the thesis of reducibility of tense. | |
From: Theodore Sider (Four Dimensionalism [2001], 4.2) |