58 ideas
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| Full Idea: Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VII.4) |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| Full Idea: A member m of M is an 'upper bound' of a subset N of M if m is not less than any member of N. A member m of M is a 'least upper bound' of N if m is an upper bound of N such that if l is any other upper bound of N, then m is less than l. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: [if you don't follow that, you'll have to keep rereading it till you do] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| Full Idea: Since combinatorial collections are enumerated, some multiplicities may be too large to be gathered into combinatorial collections. But the size of a multiplicity seems quite irrelevant to whether it forms a logical connection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| Full Idea: Many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| Full Idea: The Power Set is just he codification of the fact that the collection of functions from a mathematical collection to a mathematical collection is itself a mathematical collection that can serve as a domain of mathematical study. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| Full Idea: The Axiom of Replacement (of Skolem and Fraenkel) was remarkable for its universal acceptance, though it seemed to have no consequences except for the properties of the higher reaches of the Cantorian infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| Full Idea: The Axiom of Foundation (Zermelo 1930) says 'Every (descending) chain in which each element is a member of the previous one is of finite length'. ..This forbids circles of membership, or ungrounded sets. ..The iterative conception gives this centre stage. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.4) |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| Full Idea: Combinatorial collections (defined just by the members) obviously obey the Axiom of Choice, while it is at best dubious whether logical connections (defined by a rule) do. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| Full Idea: The controversy was not about Choice per se, but about the correct notion of function - between advocates of taking mathematics to be about arbitrary functions and advocates of taking it to be about functions given by rules. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| Full Idea: The Peano-Russell notion of class is the 'logical' notion, where each collection is associated with some kind of definition or rule that characterises the members of the collection. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.1) |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| Full Idea: The iterative conception of set was not so much as suggested, let alone advocated by anyone, until 1947. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], I) |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| Full Idea: The iterative conception of sets does not tell us how far to iterate, and so we must start with an Axiom of Infinity. It also presupposes the notion of 'transfinite iteration'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| Full Idea: The iterative conception does not provide a conception that unifies the axioms of set theory, ...and it has had very little impact on what theorems can be proved. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) | |
| A reaction: He says he would like to reject the iterative conception, but it may turn out that Foundation enables new proofs in mathematics (though it hasn't so far). |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| Full Idea: Limitation of Size has it that if a collection is the same size as a set, then it is a set. The Axiom of Replacement is characteristic of limitation of size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.5) |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
| A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| Full Idea: Mathematics is today thought of as the study of abstract structure, not the study of quantity. That point of view arose directly out of the development of the set-theoretic notion of abstract structure. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.2) | |
| A reaction: It sounds as if Structuralism, which is a controversial view in philosophy, is a fait accompli among mathematicians. |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| Full Idea: One reason to introduce the rational numbers is that it simplifes the theory of division, since every rational number is divisible by every nonzero rational number, while the analogous statement is false for the natural numbers. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.3) | |
| A reaction: That is, with rations every division operation has an answer. |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| Full Idea: The chief importance of the Continuum Hypothesis for Cantor (I believe) was that it would show that the real numbers form a set, and hence that they were encompassed by his theory. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], IV.2) |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], 2.5) |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| Full Idea: Roughly speaking, the upper and lower parts of the Dedekind cut correspond to the commensurable ratios greater than and less than a given incommensurable ratio. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], II.6) | |
| A reaction: Thus there is the problem of whether the contents of the gap are one unique thing, or many. |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| Full Idea: Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Cantor didn't mean that you could literally count the set, only in principle. |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| Full Idea: The indiscernibility of indefinitely large sizes will be a critical part of the theory of indefinitely large sizes. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| Full Idea: My proposal is that the concept of the infinite began with an extrapolation from the experience of indefinitely large size. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VIII.2) | |
| A reaction: I think it might be better to talk of an 'abstraction' than an 'extrapolition', since the latter is just more of the same, which doesn't get you to concept. Lavine spends 100 pages working out his proposal. |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| Full Idea: The intuitionist endorse the actual finite, but only the potential infinite. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.2) |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| Full Idea: The symbol 'aleph-nought' denotes the cardinal number of the set of natural numbers. The symbol 'aleph-one' denotes the next larger cardinal number. 'Aleph-omega' denotes the omega-th cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.3) |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| Full Idea: The ordinals are basic because the transfinite sets are those that can be counted, or (equivalently for Cantor), those that can be numbered by an ordinal or are well-ordered. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.4) | |
| A reaction: Lavine observes (p.55) that for Cantor 'countable' meant 'countable by God'! |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| Full Idea: The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| Full Idea: The paradox of the largest cardinal ('Cantor's Paradox') says the diagonal argument shows there is no largest cardinal, but the class of all individuals (including the classes) must be the largest cardinal number. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], III.5) |
| 22886 | The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon] |
| Full Idea: The concept of a 'limit' allows for an infinite number of finite quantities to add up to a finite sum. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 1 'Aristotle's') | |
| A reaction: This is only if the terms 'converge' on some end point. Limits are convenient fictions. |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| Full Idea: Every theorem of mathematics has a counterpart with set theory - ...but that theory cannot serve as a basis for the notion of proof. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3) |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| Full Idea: In modern mathematics virtually all work is only up to isomorphism and no one cares what the numbers or points and lines 'really are'. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
| A reaction: At least that leaves the field open for philosophers, because we do care what things really are. So should everybody else, but there is no persuading some people. |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations. | |
| From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33) |
| 22914 | An equally good question would be why there was nothing instead of something [Bardon] |
| Full Idea: If there were nothing, then wouldn't it be just as good a question to ask why there is nothing rather than something? There are many ways for there to be something, but only one way for there to be nothing. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 8 'Confronting') | |
| A reaction: [He credits Nozick with the question] I'm not sure whether there being nothing counts as a 'way' of being. If something exists it seems to need a cause, but no cause seems required for the absence of things. Nice, though. |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| Full Idea: A whole must possess an attribute peculiar to and characteristic of it as a whole; there must be a characteristic relation of dependence between the parts; and the whole must have some structure which gives it characteristics. | |
| From: Rescher,N/Oppenheim,P (Logical Analysis of Gestalt Concepts [1955], p.90), quoted by Peter Simons - Parts 9.2 | |
| A reaction: Simons says these are basically sensible conditions, and tries to fill them out. They seem a pretty good start, and I must resist the temptation to rush to borderline cases. |
| 22902 | Why does an effect require a prior event if the prior event isn't a cause? [Bardon] |
| Full Idea: To say that a reaction requires the earlier presence of an action just raises anew the question of why it is 'required' if it isn't bring about the reaction. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Pervasive') | |
| A reaction: This is another example of my demand that empiricists don't just describe and report conjunctions and patterns, but make some effort to explain them. |
| 22905 | Becoming disordered is much easier for a system than becoming ordered [Bardon] |
| Full Idea: Systems move to a higher state of entropy …because there are very many more ways for a system to be disordered than for it to be ordered. …We can also say that they tend to move from a non-equilibrium state to an equilibrium state. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Thermodynamic') | |
| A reaction: Is it actually about order, or is it just that energy radiates, and thus disperses? |
| 22913 | The universe expands, so space-time is enlarging [Bardon] |
| Full Idea: More and more space-time is literally being created from nothing all the time as the universe expands. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 8 'Realism') | |
| A reaction: [He cites Paul Davies for this] Is the universe acquiring more space, or is the given space being stretched? Acquiring more time makes no sense, so what is more space-time? |
| 22889 | We should treat time as adverbial, so we don't experience time, we experience things temporally [Bardon, by Bardon] |
| Full Idea: Kant says that instead of focusing on the nouns 'time' and 'space', it would be more on target to focus on the adverbial applications of the concepts - that we don't experience things in time and space so much as experience them temporally and spatially. | |
| From: report of Adrian Bardon (Brief History of the Philosophy of Time [2013]) by Adrian Bardon - Brief History of the Philosophy of Time 2 'Kantian' | |
| A reaction: Put like that, Kant's approach has some plausibility, given that we don't actually experience space and time as entities. To jump from that to idealism seems daft. Does every adverb imply idealism about what it specifies? |
| 22900 | How can we question the passage of time, if the question takes time to ask? [Bardon] |
| Full Idea: Even questioning the passage of time may be self-defeating: can any question be meaningfully asked or understood without presuming the passage of time from the inception of the question to its conclusion? | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Pervasive') | |
| A reaction: [He cites P.J. Zwart for this] We can at least, in B-series style, specify the starting and finishing times of the question, without talk of its passage. Nice point, though. |
| 22898 | What is time's passage relative to, and how fast does it pass? [Bardon] |
| Full Idea: If time is passing, then relative to what? How could time pass with respect to itself? Further, if time passes, at what rate does it pass? | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Pervasive') | |
| A reaction: I remember some writer grasping the nettle, and saying that time passes at one second per second. Compare travelling at one metre per metre. |
| 22897 | The A-series says a past event is becoming more past, but how can it do that? [Bardon] |
| Full Idea: In the dynamic theory of time the Battle of Waterloo is become more past. If we insist on the A-series properties, this seems inevitable. But how can a past event be changing now? | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Reasons') | |
| A reaction: [He cites Ulrich Meyer for this] We don't worry about an object changing its position when it is swept down a river. The location of the Battle of Waterloo relative to 'now' is not a property of the battle. That is a 'Cambridge' property. |
| 22901 | The B-series needs a revised view of causes, laws and explanations [Bardon] |
| Full Idea: If we accept the static (B-series) view, we have to reevaluate how we think about causation, natural laws, and scientific explanation. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Pervasive') | |
| A reaction: Any scientific account which refers to events seems to imply a dynamic view of time. Lots of scientists and philosophers endorse the static view of time, but then fail to pursue its implications. |
| 22896 | The B-series is realist about time, but idealist about its passage [Bardon] |
| Full Idea: The B-series theorist is a realist about time but an idealist about the passage of time. This is the Static Theory of time. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 4 'Reasons') | |
| A reaction: Note the both A and B are realists about time, and thus deny both the relationist and the idealist view. |
| 22903 | The B-series adds directionality when it accepts 'earlier' and 'later' [Bardon] |
| Full Idea: The static (B-series) theory, by embracing the relational temporal properties 'earlier' and 'later', adds a directional ordering to the block of events. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Time's') | |
| A reaction: I'm not clear whether this addition to the B-series picture is optional or obligatory. It is important that it seems to be a bolt-on feature, not immediately implied by the timeless series. What would Einstein say? |
| 22910 | To define time's arrow by causation, we need a timeless definition of causation [Bardon] |
| Full Idea: The problem for the causal analysis of temporal asymmetry is to come up with a definition of causation that does not itself rely on the concept of temporal asymmetry. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Causal') | |
| A reaction: This is the point at which my soul cries out 'time is a primitive concept!' Leibniz want to use dependency to define time's arrow, but how do you specify dependency if you don't know which one came first? |
| 22909 | We judge memories to be of the past because the events cause the memories [Bardon] |
| Full Idea: On the causal view of time's arrow, memories pertain to the 'past' just because they are caused by the events of which they are memories. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Causal') | |
| A reaction: How am I able to distinguish imagining the future from remembering the past? How do I tell which mental events have external causes, and which are generated by me? |
| 22904 | The psychological arrow of time is the direction from our memories to our anticipations [Bardon] |
| Full Idea: The psychological arrow of time refers to the familiar fact that that we remember (and never anticipate) the past, and anticipate (but never remember) the future. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Psychological') | |
| A reaction: Bardon rejects this on the grounds that the psychology is obviously the result of the actual order of events. Otherwise time's arrow would just result from the luck of how we individually experience things. |
| 22906 | The direction of entropy is probabilistic, not necessary, so cannot be identical to time's arrow [Bardon] |
| Full Idea: The coincidence of thermodynamic direction and the direction of time is striking, but they can't be one and the same because the thermodynamic law is merely probabilistic. Orderliness could increase, but it is highly improbable | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Thermodynamic') | |
| A reaction: This seems to be persuasive grounds for rejecting thermodynamics as the explanation of time's arrow. |
| 22907 | It is arbitrary to reverse time in a more orderly universe, but not in a sub-system of it [Bardon] |
| Full Idea: It would seem arbitrary to say that the direction of time is reversed if the whole universe becomes more orderly, but it isn't reversed for any particular sub-system that becomes more orderly. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 5 'Thermodynamic') | |
| A reaction: The thought is that if time's arrow depends on entropy, then the arrow must reverse if entropy were to reverse (however unlikely). |
| 22883 | It seems hard to understand change without understanding time first [Bardon] |
| Full Idea: It is very tough to see how we could understand what change is without understanding what time is. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], Intro) | |
| A reaction: This thought is aimed at those who are hoping to define time in terms of change. My working assumption is that time must be a primitive concept in any metaphysics. |
| 22890 | We experience static states (while walking round a house) and observe change (ship leaving dock) [Bardon] |
| Full Idea: We make a fundamental distinction between perceptions of static states and dynamic processes, …such as walking around a house, and watching a ship leave dock. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 2 'Kantian') | |
| A reaction: This seems to be a fundamental aspect of our mind, rather than of the raw experience (slightly supporting Kant). In both cases we experience a changing sequence, but we have two different interpretations of them. |
| 22884 | The motion of a thing should be a fact in the present moment [Bardon] |
| Full Idea: Whether or not something is in motion should be a fact about that thing now, not a fact about the thing in its past or in its future. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 1 'Arrow') | |
| A reaction: This is one of the present moment, in which nothing can occur if its magnitude is infinitely small. I have no solution to this problem. |
| 22892 | Experiences of motion may be overlapping, thus stretching out the experience [Bardon] |
| Full Idea: Experience itself may be constituted by overlapping, very brief, but temporally extended, acts of awareness, each of which encompassesa temporally extended streeeeetch of perceived events. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 2 'Realism') | |
| A reaction: [cites Barry Dainton 2000] I think this sounds better than Russell's suggestion, though along the same lines. I take all brain events to be a sort of memory, briefly retaining their experience. Very fast events blur because of overload. |
| 22911 | At least eternal time gives time travellers a possible destination [Bardon] |
| Full Idea: If all past, present and future events timelessly coexist, then at least there is a potential destination for the time traveller. …The Presentist treats past and future events as nonexistent, so there is no place for the time traveller to go. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 6 'Fictional') | |
| A reaction: Not a good reason to believe in the eternal block of time, of course. The growing block has a past which can be visited, but no future. |
| 22912 | Time travel is not a paradox if we include it in the eternal continuum of events [Bardon] |
| Full Idea: As long as we understand any time travel events to be timelessly included in the history of the world, and thus as part of the fixed continuum of events, time travel need not give rise to paradox. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], 6 'Time travel') | |
| A reaction: This would presumably block going back and killing your own grandparent. |
| 22882 | We use calendars for the order of events, and clocks for their passing [Bardon] |
| Full Idea: Roughly speaking, we use calendars to track the order of events in time, and clocks to track changes and the passing of events. | |
| From: Adrian Bardon (Brief History of the Philosophy of Time [2013], Intro) | |
| A reaction: So calendars cover the B-Series and clocks the A-Series, showing that this distinction is deeply embedded, and wasn't invented by McTaggart. |