Combining Texts

Ideas for 'fragments/reports', 'Philosophy of Mathematics' and 'Analogy of Religion'

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9 ideas

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
9606 The irrationality of root-2 was achieved by intellect, not experience [Brown,JR]
     Full Idea: We could not discover irrational numbers by physical measurement. The discovery of the irrationality of the square root of two was an intellectual achievement, not at all connected to sense experience.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
     A reaction: Brown declares himself a platonist, and this is clearly a key argument for him, and rather a good one. Hm. I'll get back to you on this one...
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
9612 There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
     Full Idea: A simple argument makes it clear that all mathematical arguments are abstract: there are infinitely many numbers, but only a finite number of physical entities, so most mathematical objects are non-physical. The best assumption is that they all are.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
     A reaction: This, it seems to me, is where constructivists score well (cf. Idea 9608). I don't have an infinity of bricks to build an infinity of houses, but I can imagine that the bricks just keep coming if I need them. Imagination is what is unbounded.
9610 Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
     Full Idea: Numbers are not 'abstract' (in the old sense, of universals abstracted from particulars), since each of the integers is a unique individual, a particular, not a universal.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
     A reaction: An interesting observation which I have not seen directly stated before. Compare Idea 645. I suspect that numbers should be thought of as higher-order abstractions, which don't behave like normal universals (i.e. they're not distributed).
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9620 Empiricists base numbers on objects, Platonists base them on properties [Brown,JR]
     Full Idea: Perhaps, instead of objects, numbers are associated with properties of objects. Basing them on objects is strongly empiricist and uses first-order logic, whereas the latter view is somewhat Platonistic, and uses second-order logic.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 4)
     A reaction: I don't seem to have a view on this. You can count tomatoes, or you can count red objects, or even 'instances of red'. Numbers refer to whatever can be individuated. No individuation, no arithmetic. (It's also Hume v Armstrong on laws on nature).
6. Mathematics / C. Sources of Mathematics / 7. Formalism
9639 Does some mathematics depend entirely on notation? [Brown,JR]
     Full Idea: Are there mathematical properties which can only be discovered using a particular notation?
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 6)
     A reaction: If so, this would seem to be a serious difficulty for platonists. Brown has just been exploring the mathematical theory of knots.
9629 For nomalists there are no numbers, only numerals [Brown,JR]
     Full Idea: For the instinctive nominalist in mathematics, there are no numbers, only numerals.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
     A reaction: Maybe. A numeral is a specific sign, sometimes in a specific natural language, so this seems to miss the fact that cardinality etc are features of reality, not just conventions.
9630 The most brilliant formalist was Hilbert [Brown,JR]
     Full Idea: In mathematics, the most brilliant formalist of all was Hilbert
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
     A reaction: He seems to have developed his fully formalist views later in his career. See Mathematics|Basis of Mathematic|Formalism in our thematic section. Kreisel denies that Hilbert was a true formalist.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
9608 There are no constructions for many highly desirable results in mathematics [Brown,JR]
     Full Idea: Constuctivists link truth with constructive proof, but necessarily lack constructions for many highly desirable results of classical mathematics, making their account of mathematical truth rather implausible.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
     A reaction: The tricky word here is 'desirable', which is an odd criterion for mathematical truth. Nevertheless this sounds like a good objection. How flexible might the concept of a 'construction' be?
9645 Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]
     Full Idea: If we define p as '3 if Goldbach's Conjecture is true' and '5 if Goldbach's Conjecture is false', it seems that p must be a prime number, but, amazingly, constructivists would not accept this without a proof of Goldbach's Conjecture.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 8)
     A reaction: A very similar argument structure to Schrödinger's Cat. This seems (as Brown implies) to be a devastating knock-down argument, but I'll keep an open mind for now.