Combining Philosophers

All the ideas for H.Putnam/P.Oppenheim, Richard G. Heck and William Davies

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17 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
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)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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?
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
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.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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').
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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.
14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction
20653 Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson]
     Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles.
     From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro'
     A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture?
24. Political Theory / D. Ideologies / 11. Capitalism
24071 Markets are transparent, with known prices and activity, and minimal profits [Davies,W]
     Full Idea: Markets are characterised by transparency. Prices are public, and all relevant activity is visible to everyone. And because of competition, profits are minimal, little more than a 'wage' for the seller.
     From: William Davies (Review of 'The Price is Wrong' by B.Christophers [2024], 24-04-04)
     A reaction: This account, from Braudel, is to distinguish markets from capitalism.
24073 Capitalists use their exceptional power to impose their own rules, and make the state their ally [Davies,W]
     Full Idea: Capitalists exploit their unrivalled control over time and space in order to impose their rules on everyone else. …It triumphed late, only becoming dominant in the 19th century, when it had conscripted the state as its ally.
     From: William Davies (Review of 'The Price is Wrong' by B.Christophers [2024], 24-04-04)
     A reaction: This so very much makes sense of the modern world. Nowadays capitalists are so wealthy that the state has largely become their pawn, rather than their ally. Populist leaders are their puppets (and are well rewarded).
24070 Economies have material, economic and capitalist layers [Davies,W]
     Full Idea: Braudel's economic history has three layers. At the bottom is material life of consumption, production, reproduction. Next is economic life of markets, of equals in exchange and competition. Top is capitalism, of opacity, monopoly, power, high profits.
     From: William Davies (Review of 'The Price is Wrong' by B.Christophers [2024], 24-04-04)
     A reaction: The point Davies emphasises here is the sharp distinction between the market economy and capitalism.
24074 Capitalism must mainly rely either on the labour market, or on the financial markets [Davies,W]
     Full Idea: According to Marxists, the one market capitalism cannot do without is the labour market, which creates saleable things. Others, influenced by Keynes, emphasise financial markets, where pieces of paper change hands on expectation of their value.
     From: William Davies (Review of 'The Price is Wrong' by B.Christophers [2024], 24-04-04)
     A reaction: Modern Britain fits the Keynesian account much better, given its low production, and very active (until recently) London financial market.
24072 Capitalism is the anti-market, with opacity, monopolies, powers, exceptional profits and wealth [Davies,W]
     Full Idea: Braudel sees capitalism as the 'anti-market': a world of opacity, monopoly, concentration of power and wealth, and the exceptional profits that can be achieved only by escaping the norms of 'economic life'.
     From: William Davies (Review of 'The Price is Wrong' by B.Christophers [2024], 24-04-04)
     A reaction: Given all the talk about the wonders of the 'free market' from right-wingers, this passage came as a revelation to me. Capitalists all dream of a monopoly, which is precisely the destruction of a market.