19 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. |
| 18617 | Substances, unlike aggregates, can survive a change of parts [Mumford] |
| Full Idea: Substances can survive a change in their parts in a way that a mere aggregate of parts. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 3) | |
| A reaction: A simple but very important idea. If we then distinguish between 'substances' and 'aggregates' we get a much clearer grip on things. Is the Ship of Theseus a substance or an aggregate? There is no factual answer to that. What do you want to explain? |
| 18618 | Maybe possibilities are recombinations of the existing elements of reality [Mumford] |
| Full Idea: It has been suggested that we could think of possibilities as recombinations of all the existing elements of reality. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [Armstrong 1989 is the source] The obvious problem would be that the existence of an entirely different reality would be impossible, if this was all possibility could be. It seems to cramp the style of the possible too much. Are properties elements? |
| 18619 | Combinatorial possibility has to allow all elements to be combinable, which seems unlikely [Mumford] |
| Full Idea: The combinatorial account only works if you allow that the elements are recombinable. ...But could Lincoln really have been green? It seems possible that you could jump to the moon, unless we impose some restrictions. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: Mumford suggests different combination rules for logical and natural possibility. The general objection is that combinatorial possibility is too permissive - which it clearly is. |
| 18620 | Combinatorial possibility relies on what actually exists (even over time), but there could be more [Mumford] |
| Full Idea: Can combinatorial possibility deliver enough possibilities? It uses the existing elements, but there might have been one more particular or one more property. Even extended over time, the elements seem finite, yet there could have been more. | |
| From: Stephen Mumford (Metaphysics: a very short introduction [2012], 8) | |
| A reaction: [compressed] One objection is that the theory allows too much, and now the objection is that it allows too little. Both objections are correct, so that's the end of that. But I admire the attempt to base modality on actuality. |
| 22179 | Explanatory facts also predict, and predictive facts also explain [Hempel, by Okasha] |
| Full Idea: Hempel said every scientific explanation is potentially a prediction - it would have predicted the phenomenon in question, had it not already been known. But also the information used to make a prediction is potentially an explanation. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 3 | |
| A reaction: Sounds too neatly glib to be quite true. If you explain a single event there is nothing to predict. You might predict accurately from a repetitive pattern, with no understanding at all of the pattern. |
| 6755 | For Hempel, explanations are deductive-nomological or probabilistic-statistical [Hempel, by Bird] |
| Full Idea: Hempel proposes that explanations involve covering laws and antecedent conditions; this view (the 'covering law' view) has two versions, the deductive-nomological model and the probabilistic-statistical model of explanation. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Alexander Bird - Philosophy of Science Ch.2 | |
| A reaction: The obvious problem with this approach, it seem to me, is that the laws themselves need explanation, and I don't see how a law can be foundational unless there is a divine law-giver. Are the laws arbitrary and axiomatic? |
| 17083 | The covering-law model is for scientific explanation; historical explanation is quite different [Hempel] |
| Full Idea: To put forward the covering-law models of scientific explanation is not to deny that there are other contexts in which we speak of explanation. ….That it does not fit explaining the rules of Hanoverian succession is to miss the intent of our model. | |
| From: Carl Hempel (Aspects of Scientific Explanation [1965], p. 412-3), quoted by David-Hillel Ruben - Explaining Explanation Ch 1 | |
| A reaction: Important to get that clear. It then requires a clear demarcation between science and the rest, and it had better not rule out biology because it is having a love affair with physics. |
| 13052 | Hempel rejects causation as part of explanation [Hempel, by Salmon] |
| Full Idea: Hempel explicitly rejects the idea that causality plays any essential explanatory role. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965], p.352) by Wesley Salmon - Four Decades of Scientific Explanation 1.1 | |
| A reaction: Hempel champions the 'covering-law' model of explanation. It strikes me that Hempel is so utterly wrong about this that his views aren't even a candidate for correctness, but then for a long time his views were orthodoxy. |