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17453 | The meaning of a number isn't just the numerals leading up to it |
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 |
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) |
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? |
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 |
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 |
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 |
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 |
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 |
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'). |
17449 | We can understand cardinality without the idea of one-one correspondence |
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. |
17458 | Equinumerosity is not the same concept as one-one correspondence |
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. |