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Ideas of James Robert Brown, by Text

[Canadian, fl. 1999, Professor at the University of Toronto.]

1999 Philosophy of Mathematics
Ch. 1 p.2 Mathematics is the only place where we are sure we are right
Ch. 1 p.5 The irrationality of root-2 was achieved by intellect, not experience
Ch. 1 p.5 If a proposition is false, then its negation is true
Ch. 2 p.12 There is an infinity of mathematical objects, so they can't be physical
Ch. 2 p.12 Numbers are not abstracted from particulars, because each number is a particular
Ch. 2 p.12 There are no constructions for many highly desirable results in mathematics
Ch. 2 p.12 'Abstract' nowadays means outside space and time, not concrete, not physical
Ch. 2 p.12 The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
Ch. 2 p.19 Nowadays conditions are only defined on existing sets
Ch. 2 p.19 Na´ve set theory assumed that there is a set for every condition
Ch. 2 p.22 The 'iterative' view says sets start with the empty set and build up
Ch. 3 p.40 David's 'Napoleon' is about something concrete and something abstract
Ch. 4 p.49 Empiricists base numbers on objects, Platonists base them on properties
Ch. 4 p.49 Mathematics represents the world through structurally similar models.
Ch. 4 p.53 'There are two apples' can be expressed logically, with no mention of numbers
Ch. 4 p.59 To see a structure in something, we must already have the idea of the structure
Ch. 4 p.61 Sets seem basic to mathematics, but they don't suit structuralism
Ch. 5 p.62 For nomalists there are no numbers, only numerals
Ch. 5 p.63 The most brilliant formalist was Hilbert
Ch. 5 p.65 Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
Ch. 5 p.66 Given atomism at one end, and a finite universe at the other, there are no physical infinities
Ch. 5 p.71 Berry's Paradox finds a contradiction in the naming of huge numbers
Ch. 6 p.89 Does some mathematics depend entirely on notation?
Ch. 6 p.92 A term can have not only a sense and a reference, but also a 'computational role'
Ch. 7 p.94 Definitions should be replaceable by primitives, and should not be creative
Ch. 7 p.97 A flock of birds is not a set, because a set cannot go anywhere
Ch. 7 p.102 Set theory may represent all of mathematics, without actually being mathematics
Ch. 7 p.105 When graphs are defined set-theoretically, that won't cover unlabelled graphs
Ch. 8 p.113 Constructivists say p has no value, if the value depends on Goldbach's Conjecture
Ch. 9 p.130 There is no limit to how many ways something can be proved in mathematics
Ch.10 p.154 Computers played an essential role in proving the four-colour theorem of maps
Ch.10 p.164 π is a 'transcendental' number, because it is not the solution of an equation
Ch.10 p.170 Axioms are either self-evident, or stipulations, or fallible attempts