48 ideas
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 22919 | A thing which makes no difference seems unlikely to exist [Le Poidevin] |
| Full Idea: It is a powerful argument for something's non-existence that it would make absolutely no difference. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 02 'Everything') | |
| A reaction: Powerful, but not conclusive. Neutrinos don't seem to do much, so it isn't far from there to get a particle which does nothing. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 22926 | In addition to causal explanations, they can also be inferential, or definitional, or purposive [Le Poidevin] |
| Full Idea: Not all explanations are causal. We can explain some things by showing what follows logically from what, or what is required by the definition of a term, or in terms of purpose. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 05 'Limits') | |
| A reaction: Would these fully qualify as 'explanations'? You don't explain the sea by saying that 'wet' is part of its definition. |
| 15119 | Aristotelian explanation by essence may need to draw on knowledge of other essences [Aristotle, by Koslicki] |
| Full Idea: From Aristotle's biology we learn that a successful scientific explanation of the necessary (but non-essential) features of one type of phenomenon (e.g. camels) my require appeal to facts about the essences of other types of phenomena (stomachs). | |
| From: report of Aristotle (The History of Animals [c.344 BCE]) by Kathrin Koslicki - Essence, Necessity and Explanation 13.4 |
| 22932 | We don't just describe a time as 'now' from a private viewpoint, but as a fact about the world [Le Poidevin] |
| Full Idea: In describing a time as 'now' one is not merely describing the world from one's own point of view, but describing the world as it is. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'Mystery') | |
| A reaction: If we accept this view (which implies absolute time, and the A-series view), then 'now' is not an indexical, in the way that 'I' and 'here' are indexicals. |
| 22927 | The logical properties of causation are asymmetry, transitivity and irreflexivity [Le Poidevin] |
| Full Idea: The usual logical properties of the causal relation are asymmetry (one-way), transitivity and irreflexivity (no self-causing). | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 05 'Great') | |
| A reaction: If two balls rebound off each other, that is only asymmetric if we split the action into two parts, which may be a fiction. Does a bomb cause its own destruction? |
| 22922 | We can identify unoccupied points in space, so they must exist [Le Poidevin] |
| Full Idea: If the midpoint on a line between the chair and the window is five feet from the end of the bookcase. This can be true, but if no object occupies that midpoint, then unoccupied points exist | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 03 'Lessons') | |
| A reaction: We can also locate perfect circles (running through fairy rings, or the rings of Saturn), so they must also exist. But then we can also locate the Loch Ness monster. Hm. |
| 22924 | If spatial points exist, then they must be stationary, by definition [Le Poidevin] |
| Full Idea: If there are such things as points in space, independently of any other object, then these points are by definition stationary (since to be stationary is to stay in the same place, and a point is a place). | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 03 'Search') | |
| A reaction: So what happens if the whole universe moves ten metres to the left? Is the universe defined by the objects in it (which vary), or by the space that contains them? Why can't a location move, even if that is by definition undetectable? |
| 22923 | Absolute space explains actual and potential positions, and geometrical truths [Le Poidevin] |
| Full Idea: Absolutists say space plays a number of roles. It is what we refer to when we talk of positions. It makes other things possible (by moving into unoccupied positions). And it explains geometrical truths. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 03 'Redundancy') | |
| A reaction: I am persuaded by these, and am happy to treat space (and time) as a primitive of metaphysics. |
| 22928 | For relationists moving an object beyond the edge of space creates new space [Le Poidevin] |
| Full Idea: For the relationist, if Archytas goes to the edge of space and extends his arm, he is creating a new spatial relation between objects, and thus extending space, which is, after all, just the collection of thos relations. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 05 'beyond') | |
| A reaction: The obvious point is what are you moving your arm into? And how can some movements be in space, while others create new space? It's a bad theory. |
| 22931 | We distinguish time from space, because it passes, and it has a unique present moment [Le Poidevin] |
| Full Idea: The most characteristic features of time, which distinguish it from space, are the fact that time passes, and the fact that the present is in some sense unique | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'Mystery') | |
| A reaction: The B-series view tries to avoid passing time and present moments. I suspect that modern proponents of the B-series mainly want to unifying their view of time with Einstein's, to give us a scientific space-time. |
| 22917 | Since nothing occurs in a temporal vacuum, there is no way to measure its length [Le Poidevin] |
| Full Idea: Since, by definition, nothing happens in a temporal vacuum, there is no possible means of determining its length. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 02 'without change') | |
| A reaction: This is offered a part of a dubious proof that a temporal vacuum is impossible. I like Shoemaker's three worlds thought experiment, which tests this idea to the limit. |
| 22921 | Temporal vacuums would be unexperienced, unmeasured, and unending [Le Poidevin] |
| Full Idea: Three arguments that a temporal vacuum is impossible: we can't experience it, we can't measure it, and it would have no reason to ever terminate. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 03 'Lessons') | |
| A reaction: [summarised] The first two reasons are unimpressive. The interiors of black holes are off limits for us. The arrival of time into a timeless situation may actually have occurred, but be beyond our understanding. |
| 22934 | Time can't speed up or slow down, so it doesn't seem to be a 'process' [Le Poidevin] |
| Full Idea: Processes can speed up or slow down, but surely the passage of time is not something that can speed up or slow down? | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'Mystery') | |
| A reaction: If something is a process we can ask 'process of what?', but the only answer seems to be that it's a process of processing. So it is that which makes processes possible (and so, as I keep saying) it is best viewed as a primitive. |
| 22938 | To say that the past causes the present needs them both to be equally real [Le Poidevin] |
| Full Idea: The causal connection between the past and the present seems to require that the past is as real as the present. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'First') | |
| A reaction: Cause and effect need to conjoin in space, but their subsequent separation doesn't seem to be a problem. The idea that causes and their effects must be eternally compresent is an absurdity. |
| 22939 | The B-series doesn't seem to allow change [Le Poidevin] |
| Full Idea: How can anything change in a B-universe? | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'Second') | |
| A reaction: It seems that change needs time to move on. A timeless series of varying states doesn't seem to be the same thing as change. B-seriesers must be tempted to deny change, and yet nothing seems more obvious to us than change. |
| 22940 | If the B-universe is eternal, why am I trapped in a changing moment of it? [Le Poidevin] |
| Full Idea: What in the B-universe determines my temporal perspective? I can move around in space at will, but I have no choice over where I am in time. What time I am is something that changes, and again I have no control over that | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'Second') | |
| A reaction: The B-series always has to be asserted from the point of view of eternity (e.g. by Einstein). Yet an omniscient mind would still see each of us trapped in our transient moments, so that is part of eternal reality. |
| 22947 | An ordered series can be undirected, but time favours moving from earlier to later [Le Poidevin] |
| Full Idea: A series can be ordered without being directed (such as the series of integers), …but the passage of time indicates a preferred direction, moving from earlier to later events, and never the other way around. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'Hidden') | |
| A reaction: I wonder what 'preferred' means here? It is not just memory versus anticipation. The saddest words in the English language are 'Too late!'. It is absurd to say that being too late is an illusion. |
| 22952 | If time's arrow is causal, how can there be non-simultaneous events that are causally unconnected? [Le Poidevin] |
| Full Idea: An objection to the Causal analysis of time's arrow is that it is surely possible for non-simultaneous events to be causally unconnected. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'Seeds') | |
| A reaction: I suppose the events could be linked causally by intermediaries. If reality is a vast causal nexus, everything leads to everything else, in some remote way. It's still a good objections, though. |
| 22951 | If time's arrow is psychological then different minds can impose different orders on events [Le Poidevin] |
| Full Idea: If the Psychological account of time's arrow is correct …then there is nothing to prevent different minds from imposing different orders on the world. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'The mind's') | |
| A reaction: All we need is for two people to disagree about the order of some past events. The idea that we are psychologically creating time's arrow when everyone feels they are its victims strikes me as a particularly silly theory. |
| 22948 | There are Thermodynamic, Psychological and Causal arrows of time [Le Poidevin] |
| Full Idea: The three most significant arrows of time are the Thermodynamic (the direction from order to disorder), the Psychological (from perceptions of events to memories), and the Causal (from cause to effect). | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'Three') | |
| A reaction: It would be nice if one of these explained the other two. Le Poidevin rejects the Psychological arrow, and seems to favour the Causal. Since I favour taking time as a primitive, I'm inclined to think that the arrow is included in the deal. |
| 22949 | Presumably if time's arrow is thermodynamic then time ends when entropy is complete [Le Poidevin] |
| Full Idea: One consequence of the Thermodynamic analysis of time's arrow is that a universe in which things are as disordered as they could be would exhibit no direction of time at all, because there would be no more significant changes in entropy. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'Three') | |
| A reaction: And presumably time would gradually fizzle out, rather than ending abruptly. If entropy then went into reverse, there would be no time interval between the end and the new beginning. Entropy can vary locally, so it has to be universal. |
| 22950 | If time is thermodynamic then entropy is necessary - but the theory says it is probable [Le Poidevin] |
| Full Idea: The Second Law of Thermodynamics says it is overwhelmingly probable that entropy will increase. This leaves the door open for occasional isolated instances of decrease. But the thermodynamic arrow makes the increase a necessity. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'Three') | |
| A reaction: Le Poidevin sees this as a clincher against the thermodynamic explanation of the arrow. I'm now sure how the Second Law can even be stated without explicit or implicit reference to time. |
| 22953 | Time's arrow is not causal if there is no temporal gap between cause and effect [Le Poidevin] |
| Full Idea: If there is no temporal gap between cause and effect, then the causal analysis of time's arrow is doomed. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 12 'simultaneous') | |
| A reaction: A number of recent commentators have rejected the sharp distinction between cause and effect, seeing it as a unified process (which takes time to occur). |
| 22943 | Instantaneous motion is an intrinsic disposition to be elsewhere [Le Poidevin] |
| Full Idea: Being in motion at a particular time can be an intrinsic property of an object, as a disposition to be elsewhere than the place it is. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 09 'in present') | |
| A reaction: This needs an ontology which includes unrealised dispositions. People trapped in boring meetings have a disposition to be elsewhere, but they are stuck. I think 'power' is a better word here than 'disposition'. The disposition isn't just for 'elsewhere'. |
| 22945 | The dynamic view of motion says it is primitive, and not reducible to objects, properties and times [Le Poidevin] |
| Full Idea: According to the dynamic account of motion, an object's being in motion is a primitive event, not further analysable in terms of objects, properties and times. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 09 'Zeno') | |
| A reaction: [The rival view is 'static'] Physics suggests that motion may be indefinable, but acceleration can be given a reductive account. If time and space are taken as primitive (which seems sensible to me), then making motion also primitive is a bit greedy. |
| 22937 | If the present could have diverse pasts, then past truths can't have present truthmakers [Le Poidevin] |
| Full Idea: If any number of pasts are compatible with the present state of affairs, and it is only the present state of affairs which can make true or false statements about the past, then no statement about the past is either true or false. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 08 'First') | |
| A reaction: He suggests an explosion which could have had innumerable different causes. The explosion could have had different origins, but not sure that the whole of present reality could. Presentists certainly have problems with truthmakers for the past. |
| 22925 | The present is the past/future boundary, so the first moment of time was not present [Le Poidevin] |
| Full Idea: The present is the boundary between past and future, therefore if there was a first moment of time, it could not have been present - because there can be no past at the beginning of time. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 05 'Limits') | |
| A reaction: How about at the start of a race the athletes cannot be running. How about 'all moments of time have preceding moments - apart from the first moment'? |
| 22944 | The primitive parts of time are intervals, not instants [Le Poidevin] |
| Full Idea: Intervals of time can be viewed as primitive, and not decomposable into a series of instants. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 09 'in present') | |
| A reaction: Given that instants are nothing, and intervals are something, the latter are clearly the better candidates to be the parts of time. Is there a smallest interval? |
| 22942 | If time is infinitely divisible, then the present must be infinitely short [Le Poidevin] |
| Full Idea: Assuming time to be infinitely divisible, the present can have no duration at all, for if it did, we could divide it into parts, and some parts would be earlier than others. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 09 'in present') | |
| A reaction: I quite like Aristotle's view that things only have parts when you actually divide them. In modern physics fields don't seem to be infinitely divisible. It's a puzzle, though, innit? |
| 22946 | The multiverse is distinct time-series, as well as spaces [Le Poidevin] |
| Full Idea: The multiverse is not just a collection of distinct spaces, it is also a collection of distinct time-series. | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 11 'Objections') | |
| A reaction: This boggles the imagination even more than distinct spatial universes. |
| 23302 | Plants have far less life than animals, but more life than other corporeal entities [Aristotle] |
| Full Idea: The genus of plants, whilst it is devoid of life compared with an animal, is endowed with life as compared with other corporeal entities. In the sea there are certain objects which one would be at a loss to determine whether they be animal or vegetable. | |
| From: Aristotle (The History of Animals [c.344 BCE], 588b09) | |
| A reaction: It seems that Aristotle takes life to come in degrees, assessed by the amount of physical vitality observed. This seems to make lambs more alive than sheep, which isn't very plausible. This is part of his 'gradualist' view of nature. |
| 23301 | There is a gradual proceeding from the inanimate to animals, with no clear borderlines [Aristotle] |
| Full Idea: Nature proceeds little by little from things lifeless to animal life so that it is impossible to determine the exact line of demarcation, nor on which side an intermediate form should lie. ...In plants there is a continuous ascent towards the animal. | |
| From: Aristotle (The History of Animals [c.344 BCE], 588b04) | |
| A reaction: This in itself should have alerted medieval Christians to the problematic nature of the idea that animal species were divinely created. |
| 22941 | How could a timeless God know what time it is? So could God be both timeless and omniscient? [Le Poidevin] |
| Full Idea: Could a timeless being now know what the time was? If so, does this show that there must be something wrong with the idea of God as both timeless and omniscient? | |
| From: Robin Le Poidevin (Travels in Four Dimensions [2003], 09 'Questions') | |
| A reaction: This is a potential contradiction between the perfections of a supreme God which I had not noticed before. Leibniz tried to refute such objections, but not very successfully, I think. |