20 ideas
| 17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
| Full Idea: My knowing what the number '33' denotes cannot consist in my knowing that it denotes the number of decimal numbers between '1' and '33', because I would know that even if it were in hexadecimal (which I don't know well). | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5) | |
| A reaction: Obviously you wouldn't understand '33' if you didn't understand what '33 things' meant. |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| Full Idea: An appreciation of the connection between sameness of number and equinumerosity that it reports is essential to even the most primitive grasp of the concept of cardinal number. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6) |
| 17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
| Full Idea: One need not conceive of the numerals as objects in their own right in order to count. The numerals are not mentioned in counting (as objects to be correlated with baseball players), but are used. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 3) | |
| A reaction: He observes that when you name the team, you aren't correlating a list of names with the players. I could correlate any old tags with some objects, and you could tell me the cardinality denoted by the last tag. I do ordinals, you do cardinals. |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| Full Idea: I am not denying that counting can be done mindlessly, without making judgments of cardinality along the way. ...But the question is whether counting is, as it were, fundamentally a mindless exercise. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5) | |
| A reaction: He says no. It seems to me like going on a journey, where you can forget where you are going and where you have got to so far, but those underlying facts are always there. If you just tag things with unknown foreign numbers, you aren't really counting. |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
| Full Idea: Counting is not mere tagging: it is the successive assignment of cardinal numbers to increasingly large collections of objects. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5) | |
| A reaction: That the cardinals are 'successive' seems to mean that they are ordinals as well. If you don't know that 'seven' means a cardinality, as well as 'successor of six', you haven't understood it. Days of the week have successors. Does PA capture cardinality? |
| 17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
| Full Idea: It is far from obvious that knowing what 'just as many' means requires knowing what a one-one correspondence is. The notion of a one-one correspondence is very sophisticated, and it is far from clear that five-year-olds have any grasp of it. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 4) | |
| A reaction: The point is that children decide 'just as many' by counting each group and arriving at the same numeral, not by matching up. He cites psychological research by Gelman and Galistel. |
| 17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |
| Full Idea: 'Just as many' is independent of the ability to count, and we shouldn't characterise equinumerosity through counting. It is also independent of the concept of number. Enough cookies to go round doesn't need how many cookies. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 4) | |
| A reaction: [compressed] He talks of children having an 'operational' ability which is independent of these more sophisticated concepts. Interesting. You see how early man could relate 'how many' prior to the development of numbers. |
| 17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |
| Full Idea: The interest of Frege's Theorem is that it offers us an explanation of the fact that the numbers satisfy the Dedekind-Peano axioms. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6) | |
| A reaction: He says 'explaining' does not make it more fundamental, since all proofs explain why their conclusions hold. |
| 17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
| Full Idea: For a long time my daughter had no understanding of the question of how many numerals or numbers there are between 'one' and 'five'. I think she lacked the concept of numerals as objects which can themselves be counted. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 5) | |
| A reaction: I can't make any sense of numbers actually being objects, though clearly treating all sorts of things as objects helps thinking (as in 'the victory is all that matters'). |
| 17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
| Full Idea: Equinumerosity is not the same concept as being in one-one correspondence with. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 6) | |
| A reaction: He says this is the case, even if they are coextensive, like renate and cordate. You can see that five loaves are equinumerous with five fishes, without doing a one-one matchup. |
| 17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
| Full Idea: One can have a perfectly serviceable concept of cardinality without so much as having the concept of one-one correspondence. | |
| From: Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 3) | |
| A reaction: This is the culmination of a lengthy discussion. It includes citations about the psychology of children's counting. Cardinality needs one group of things, and 1-1 needs two groups. |
| 1556 | By nature people are close to one another, but culture drives them apart [Hippias] |
| Full Idea: I regard you all as relatives - by nature, not by convention. By nature like is akin to like, but convention is a tyrant over humankind and often constrains people to act contrary to nature. | |
| From: Hippias (fragments/reports [c.430 BCE]), quoted by Plato - Protagoras 337c8 |
| 21332 | We don't get a love of 'order' from nature - which is thoroughly chaotic [Mill] |
| Full Idea: Even the love of 'order' which is thought to be a following of the ways of nature is in fact a contradiction of them. All which people are accustomed to deprecate as 'disorder' is precisely a counterpart of nature's ways. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: The Greeks elevated the idea that the cosmos was orderly, but almost entirely based on the regular movement of the planets. They turned a blind eye to the messy bits of nature. As you magnify nature, order and chaos seem to alternate. |
| 21333 | Evil comes from good just as often as good comes from evil [Mill] |
| Full Idea: If good frequently comes out of evil, the converse fact, evil coming out of good, is equally common. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.117) | |
| A reaction: Mill surmises that on the whole good comes from good, and evil from evil, but the point is that the evidence doesn't favour the production of increased good. |
| 21335 | Belief that an afterlife is required for justice is an admission that this life is very unjust [Mill] |
| Full Idea: The necessity of redressing the balance [of injustice] is deemed one of the strongest arguments for another life after death, which amounts to an admission that the order of things in this life is often an example of injustice, not justice. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874]) | |
| A reaction: It certainly seems that an omnipotent God could administer swift justice in this life. If the whole point is that we need freedom of will, then why is justice administered at a much later date? The freedom seems to be illusory. |
| 21334 | No necessity ties an omnipotent Creator, so he evidently wills human misery [Mill] |
| Full Idea: If a Creator is assumed to be omnipotent, if he bends to a supposed necessity, he himself makes the necessity which he bends to. If the maker of the world can all that he will, he wills misery, and there is no escape from the conclusion. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.119) | |
| A reaction: If you add that the Creator is supposed to be perfectly benevolent, you arrive at the paradox which Mackie spells out. Is the correct conclusion that God exists, and is malevolent? Mill doesn't take that option seriously. |
| 21329 | Nature dispenses cruelty with no concern for either mercy or justice [Mill] |
| Full Idea: All of this [cruel killing] nature does with the most supercilious disregard both of mercy and of justice, emptying her shafts upon the best and noblest indifferently with the meanest and worst | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: The existence of an afterlife at least offers an opportunity to rectify any injustice, but that hardly meets the question of why there was injustice in the first place. It would be odd if it actually is justice, but none of us can see why that is so. |
| 21328 | Killing is a human crime, but nature kills everyone, and often with great tortures [Mill] |
| Full Idea: Killing, the most criminal act recognised by human laws, nature does once to every being that lives, and frequently after protracted tortures such as the greatest know monsters purposely inflicted on their living fellow creatures | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: We certainly don't condemn lions for savaging gazelles, but the concept of a supreme mind controlling nature forces the question. Theology needs consistency between human and divine morality, and the supposed derivation of the former from the latter. |
| 21330 | Nature makes childbirth a miserable experience, often leading to the death of the mother [Mill] |
| Full Idea: In the clumsy provision which nature has made for the perpetual renewal of animal life, ...no human being ever comes into the world but another human being is literally stretched on the rack for hours or day, not unfrequently issuing in death. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: This is a very powerful example, which is rarely cited in modern discussions. |
| 21331 | Hurricanes, locusts, floods and blight can starve a million people to death [Mill] |
| Full Idea: Nature often takes the means by which we live. A single hurricane, a flight of locusts, or an inundation, or a trifling chemical change in an edible root, starve a million people. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: [second sentence compressed] The 'edible root' is an obvious reference to the Irish potato famine. Some desertification had human causes, but these are telling examples. |