43 ideas
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
| A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
| Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
| A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
| 15682 | Even fairly simple animals make judgements based on categories [Gelman] |
| Full Idea: All organisms form categories: even mealworms have category-based preferences, and higher-order animals such as pigeons or octopi can display quite sophisticated categorical judgements. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Prelims') | |
| A reaction: [She cites some 1980 research to support this] This comes as no surprise, as I take categorisation as almost definitive of what a mind is. My surmise is that some sort of 'labelling' system is at the heart of it (like Googlemail labels!). |
| 15691 | Children accept real stable categories, with nonobvious potential that gives causal explanations [Gelman] |
| Full Idea: By five children assume that a variety of categories have rich inductive potential, are stable over outward transformations, include crucial nonobvious properties, have innate potential, privilege causal features, can be explained causally, and are real. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'Intro') | |
| A reaction: This is Gelman's helpful summary of the findings of research on childhood essentialising, and says the case for this phenomenon is 'compelling'. |
| 15700 | In India, upper-castes essentialize caste more than lower-castes do [Gelman] |
| Full Idea: The notion of caste in India is more essentialized among upper-caste than lower-caste individuals. | |
| From: Susan A. Gelman (The Essential Child [2003], 08 'Intro') | |
| A reaction: In a book defending fairly innate essentialism in the human race, Gelman offers this point as a warning that large cultural ingredients can be involved. Racism is the classic difficulty with essentialism. |
| 15685 | Essentialism is either natural to us, or an accident of our culture, or a necessary result of language [Gelman] |
| Full Idea: The two views contrasting with essentialism naturally emerging in childhood are the claim that essentialism is a historical accident emerging from Western philosophy, and that essentialism is an inherent consequence of naming things. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Background') | |
| A reaction: Helpful. I take Idea 15682 to rule out the idea that it is just a feature of western culture. I can't conceive of early man surviving without essentialism. I don't think it rules out the naming view. Animals may do what emerges in us as full 'naming'. |
| 15684 | Children's concepts include nonobvious features, like internal parts, functions and causes [Gelman] |
| Full Idea: Children incorporate a variety of nonobvious features into their concepts, including internal parts, functions, causes, and ontological distinctions. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Prelims') | |
| A reaction: This remark sums up the general thesis of her book, which she supports with a wealth of first-hand evidence. It supports my view, that the desire and need for explanation is at the root of essentialist concepts. It's hard wired in us. |
| 15681 | Essentialism: real or representational? sortal, causal or ideal? real particulars, or placeholders? [Gelman] |
| Full Idea: We map types of essentialism by asking is it in the world or in our representations, is it sortal or causal or ideal, and is it specific particulars or placeholders for the unknown? | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Prelims') | |
| A reaction: I am struck by the way that this practising experimental psychologist gets to ask questions and make distinctions much more extensively than most armchair philosophers on the subject. She focuses on the representational, causal, placeholder view. |
| 15678 | Essentialism says categories have a true hidden nature which gives an object its identity [Gelman] |
| Full Idea: Essentialism is the view that categories have an underlying reality or true nature that one cannot observe directly but that gives an object its identity. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Intro') | |
| A reaction: I think the introduction of categories here is a misunderstanding. Does an uncategorisable thing therefore have no identity (even though it has properties)? If categories give objects their identity, what gives categories their identity? |
| 15683 | Sortals are needed for determining essence - the thing must be categorised first [Gelman] |
| Full Idea: I suggest that sortals are likewise required for determining essence. One cannot answer the question 'What is the essence of this?' without supplying the sortal - of this 'what'. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Prelims') | |
| A reaction: I remain baffled by this view. I take the category to be an inductive generalisation from other similar individuals. It can't get off the ground if you don't start with the individuals. Sortals are just a shorthand. |
| 15697 | Kind (unlike individual) essentialism assumes preexisting natural categories [Gelman] |
| Full Idea: With kind essentialism the person assumes that the world is divided up into preexisting natural categories. Individual essentialism seems not to require any such commitment to kind realism. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'Essentialism') | |
| A reaction: This pinpoints my difficulty: how do we decide whether some category or attributed essence is part of a preexisting natural kind? Some natural kinds are self-evident, like water (roughly), but others need subtle teasing out. How is the teasing done? |
| 15687 | Kinship is essence that comes in degrees, and age groups are essences that change over time [Gelman] |
| Full Idea: Kinship is essentialized, but admits of degrees, ...and people can be essentialist even about categories they do not view as fixed over time, such as age groupings. | |
| From: Susan A. Gelman (The Essential Child [2003], 03 'Summary') | |
| A reaction: Given my notion of essence are necessarily explanatory, I embrace both of these points. Being very athletic comes in degrees, and changes over times. |
| 15679 | Essentialism comes from the cognitive need to categorise [Gelman] |
| Full Idea: Essentialism has its source in the cognitive requirement of categorization in certain domains - particularly as they affect the young learner. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Essentialist') | |
| A reaction: I think the phenomenon is better understood as part of the cognitive requirement to understand and explain. Categorisation is just one way to aid explanation. Children try to understand (essentially) a new animal without categorisation. |
| 15698 | We found no evidence that mothers teach essentialism to their children [Gelman] |
| Full Idea: We found no evidence that mothers teach essentialism to their children. ...Mothers teach children about kinds, not about essences, and mothers help children identify which categories are richly structured. | |
| From: Susan A. Gelman (The Essential Child [2003], 07 'Conclusions') | |
| A reaction: This is a psychologist who specialises in this topic. If you think essentialism is inculcated by a our culture, you will have to blame the fathers. |
| 15709 | Essentialism is useful for predictions, but it is not the actual structure of reality [Gelman] |
| Full Idea: Essentialism is a reasoning heuristic that allows us to make fairly good predictions much of the time, but it should not be confused with the structure of reality. | |
| From: Susan A. Gelman (The Essential Child [2003], 11 'Discussion') | |
| A reaction: She particularly cites biology as the area where it might be inaccurate. I'm beginning to think that the operations of induction are the place to look for an good understanding of essentialism. |
| 15696 | Peope favor historical paths over outward properties when determining what something is [Gelman] |
| Full Idea: People favor historical paths over outward properties when determining what something is. ...An object looking like a knife is less likely to be called 'a knife' if it is described as having been created by accident. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'Essentialism') | |
| A reaction: I like this because it talks, suggestively, of 'historical paths' rather than of 'origin'. Thus we might judge a person's identity by their traumatic experience rather than by their birth. This doesn't challenge necessity of origin, but affects labels. |
| 15707 | There is intentional, mechanical, teleological, essentialist, vitalist and deontological understanding [Gelman] |
| Full Idea: The modes of understanding (or modes of construal) which have been proposed are intentional, mechanical, teleological, essentialist, vitalist (perhaps), and deontological. | |
| From: Susan A. Gelman (The Essential Child [2003], 11 'Broadening') | |
| A reaction: She cites psychological research to support this, and calls it 'a relatively small number' of modes. Compare Aristotle's four modes of cause/explanation. |
| 15703 | Memories often conform to a theory, rather than being neutral [Gelman] |
| Full Idea: Memory is notorious for conforming to theory (rather than memory being a neutral source of information). | |
| From: Susan A. Gelman (The Essential Child [2003], 09 'Theory') | |
| A reaction: This observation by a psychologist is music to sceptics about objectivity. Memory is so fundamental to our basic epistemology that it could even be the nature of thought itself. |
| 15708 | Inductive success is rewarded with more induction [Gelman] |
| Full Idea: Inductive success is rewarded with more induction. | |
| From: Susan A. Gelman (The Essential Child [2003], 11 'Broadening') | |
| A reaction: I love this one. Neat, accurate, and central to how we understand the world. I take inductive success to be stored as labels, concepts, categories, words and general truths, which are then our resource for further attempts. |
| 15695 | Children make errors in induction by focusing too much on categories [Gelman] |
| Full Idea: Because of their narrow focus, children's sensitivity to categories as the basis of induction is a reasoning bias that, though useful much of the time, results in systematic errors. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'The role') | |
| A reaction: This is the bad sense of 'essentialism' which worries its opponents. Presumably, though, my favoured scientific essentialism will be 'scientific', and avoid this problem. The relation between categories and induction needs to be clear. |
| 15694 | Children overestimate the power of a single example [Gelman] |
| Full Idea: We suggest that children overestimate the power of a single example. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'The role') | |
| A reaction: This conclusion arises from extensive psychological research. 'My grandma smoked, and she lived to be 97' - adults do this too. Wittgenstein says assuming other minds because of your own is induction from one example! |
| 15692 | People tend to be satisfied with shallow explanations [Gelman] |
| Full Idea: People tend to be satisfied with rather shallow explanations. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'Is essentialism') | |
| A reaction: She cites some psychological research to support this. Pretty obvious really. I take the so-called 'scientific method' to be nothing more than ceasing to be satisfied with such shallowness. |
| 15680 | Folk essentialism rests on belief in natural kinds, in hidden properties, and on words indicating structures [Gelman] |
| Full Idea: The three components of essentialism as a folk belief are the idea that certain categories are natural kinds, the idea that some unobservable property causes the way things are, and the idea that words reflect real structures. | |
| From: Susan A. Gelman (The Essential Child [2003], 01 'Prelims') |
| 15686 | Labels may indicate categories which embody an essence [Gelman] |
| Full Idea: Labels may signal categories that are believed to embody an essence. | |
| From: Susan A. Gelman (The Essential Child [2003], 02 'Privileged') | |
| A reaction: This is quoted by her, as a summary of a substantial body of research which she endorses. I cite it because it pinpoints my own view. I take 'labels' to be basic to minds, as organisers of thought, and this ties essences to labels. Satisfying picture. |
| 15690 | Causal properties are seen as more central to category concepts [Gelman] |
| Full Idea: Properties that enter into causally meaningful links are better remembered and are treated as more central to the category than properties that are not causally meaningful. | |
| From: Susan A. Gelman (The Essential Child [2003], 05 'Causation2') | |
| A reaction: This is a summary of considerable psychological research. This account not only sounds plausible, but would fit better withy why we form concepts and categories in the first place. We are trying to relate to the causations of nature. |
| 15688 | Categories are characterized by distance from a prototype [Gelman] |
| Full Idea: On prototype views, categories are characterized by distance from a prototype. | |
| From: Susan A. Gelman (The Essential Child [2003], 05 'Causation') | |
| A reaction: Gelman observes that this view makes no reference to any causal features of things. This cuts them off from using underlying essences in the process of categorisation and concept-formation. How do you spot a prototype, with no category? |
| 15689 | Theory-based concepts use rich models to show which similarities really matter [Gelman] |
| Full Idea: Theory-based approaches to categories are a response to the limitations of mere similarities holding the category together, and require knowledge-rich explanatory models to say which features are more central to a concept. | |
| From: Susan A. Gelman (The Essential Child [2003], 05 'Causation1') | |
| A reaction: I see a promising account in linking theory theory to essentialism. For a physical object (or even for a process) infer a structure, and then identify what is most important in that structure. That gives you your stable, agreed concept. |
| 15699 | Prelinguistic infants acquire and use many categories [Gelman] |
| Full Idea: Language does not appear to be necessary for forming categories, since prelinguistic infants acquire many categories, and even use categories to form inferences about unknown properties. | |
| From: Susan A. Gelman (The Essential Child [2003], 08 'Intro') | |
| A reaction: She cites lots of research in support of this claim. The idea may come as a surprise to some people, but not to me. I take it that categorisation is what a brain is for, including animal brains. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 15693 | One sample of gold is enough, but one tree doesn't give the height of trees [Gelman] |
| Full Idea: We can confidently determine the chemical composition of gold from just a single sample, but we cannot determine the height of trees from just a single tree. | |
| From: Susan A. Gelman (The Essential Child [2003], 06 'The role') | |
| A reaction: The tricky word here is 'confidently'. If you meet one Latvian who is nice, do you assume they are all nice? At what point do you decide gold etc. really are natural kinds, where one sample tells all? Evolution of species... |
| 15701 | Nouns seem to invoke stable kinds more than predicates do [Gelman] |
| Full Idea: Children judged personal characteristics as more stable when they were referred to by a noun ('She is a carrot eater') than by a verbal predicate ('She eats carrots whenever she can') | |
| From: Susan A. Gelman (The Essential Child [2003], 08 'Naming') | |
| A reaction: This fits with my feeling that 'labels' are the basis of how the mind works. The noun invokes a genuine category of thing, where a predicate attaches to some preselected category ('she'). Gelman says names encourage inductions. |
| 15702 | Essentialism doesn't mean we know the essences [Gelman] |
| Full Idea: Essentialism does not entail that people know what the essence is. | |
| From: Susan A. Gelman (The Essential Child [2003], 09 'Theory') | |
| A reaction: This is a fundamental and (I would say) fairly obvious point, but it needs to be made to the more passionate opponents of essentialism. |
| 15705 | Essentialism encourages us to think about the world scientifically [Gelman] |
| Full Idea: Essentialism encourages a 'scientific' mindset in thinking about the natural world, a belief that intensive study of a natural domain will yield ever more underlying properties. | |
| From: Susan A. Gelman (The Essential Child [2003], 11 'Intro') | |
| A reaction: Maybe scientists must be committed to essences, the way mathematicians must be committed to numbers? This idea spendidly opposes the doubts expressed by Popper. |
| 15704 | Essentialism starts from richly structured categories, leading to a search for underlying properties [Gelman] |
| Full Idea: If my speculations are correct, then essentialism starts out strictly as a belief that many categories are richly structured kinds, then additionally becomes a search for underlying inherent properties. | |
| From: Susan A. Gelman (The Essential Child [2003], 10 'Figuring') | |
| A reaction: This is her summary of extensive essentialist research among children. She favours the priority of kinds and categories. We actually change taxonomies on the basis of revisions in our accounts of essence. Science negotiates. |
| 15706 | A major objection to real essences is the essentialising of social categories like race, caste and occupation [Gelman] |
| Full Idea: One major argument against the view that essences are real is the rampant essentializing of categories that are socially constructed (such as race, caste and occupation). | |
| From: Susan A. Gelman (The Essential Child [2003], 11 'Is essentialism') | |
| A reaction: You can't argue with that. It raises the question of whether the approach of scientific essentialism has any value in the social, rather than physical, sciences. We jokingly essentialise groups of people such as referees or Oxonians. |