| 15916 | Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine] |
| Full Idea: Frege assumed that since infinite collections cannot be counted, he needed a theory of number that is independent of counting. He therefore took one-to-one correspondence to be basic, not well-orderings. Hence cardinals are basic, not ordinals. | |
| From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 17446 | Counting rests on one-one correspondence, of numerals to objects [Frege] |
| Full Idea: Counting rests itself on a one-one correlation, namely of numerals 1 to n and the objects. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894]), quoted by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: Parsons observes that counting will establish a one-one correspondence, but that doesn't make it the aim of counting, and so Frege hasn't answered Husserl properly. Which of the two is conceptually prior? How do you decide. |
| 9582 | Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege] |
| Full Idea: When Husserl says that sameness of number can be shown by one-one correlation, he forgets that this counting itself rests on a univocal one-one correlation, namely that between the numerals 1 to n and the objects of the set. | |
| From: Gottlob Frege (Review of Husserl's 'Phil of Arithmetic' [1894], p.326) | |
| A reaction: This is the platonist talking. Neo-logicism is attempting to build numbers just from the one-one correlation of objects. |
| 17444 | Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck] |
| Full Idea: Husserl famously argued that one should not explain number in terms of equinumerosity (or one-one correspondence), but should explain equinumerosity in terms of sameness of number, which should be characterised in terms of counting. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: [Heck admits he hasn't read the Husserl] I'm very sympathetic to Husserl, though nearly all modern thinking favours Frege. Counting connects numbers to their roots in the world. Mathematicians seem oblivious of such things. |
| 14118 | We can define one-to-one without mentioning unity [Russell] |
| Full Idea: It is possible, without the notion of unity, to define what is meant by one-to-one. | |
| From: Bertrand Russell (The Principles of Mathematics [1903], §109) | |
| A reaction: This is the trick which enables the Greek account of numbers, based on units, to be abandoned. But when you have arranged the boys and the girls one-to-one, you have not yet got a concept of number. |
| 9852 | We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett] |
| Full Idea: A child understands 'there are just as many nuts as apples' as easily by pairing them off as by counting them. | |
| From: Michael Dummett (Frege philosophy of mathematics [1991], Ch.12) | |
| A reaction: I find it very intriguing that you could know that two sets have the same number, without knowing any numbers. Is it like knowing two foreigners spoke the same words, without understanding them? Or is 'equinumerous' conceptually prior to 'number'? |
| 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. |