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Ideas of James Robert Brown, by Text
[Canadian, fl. 1999, Professor at the University of Toronto.]
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1999
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Philosophy of Mathematics
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Ch. 1
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p.2
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9604
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Mathematics is the only place where we are sure we are right
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Ch. 1
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p.5
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9605
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If a proposition is false, then its negation is true
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Ch. 1
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p.5
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9606
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The irrationality of root-2 was achieved by intellect, not experience
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Ch. 2
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p.12
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9612
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There is an infinity of mathematical objects, so they can't be physical
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Ch. 2
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p.12
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9610
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Numbers are not abstracted from particulars, because each number is a particular
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Ch. 2
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p.12
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9608
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There are no constructions for many highly desirable results in mathematics
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Ch. 2
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p.12
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9611
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'Abstract' nowadays means outside space and time, not concrete, not physical
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Ch. 2
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p.12
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9609
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The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
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Ch. 2
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p.19
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9613
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Naïve set theory assumed that there is a set for every condition
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Ch. 2
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p.19
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9615
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Nowadays conditions are only defined on existing sets
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Ch. 2
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p.22
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9617
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The 'iterative' view says sets start with the empty set and build up
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Ch. 3
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p.40
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9619
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David's 'Napoleon' is about something concrete and something abstract
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Ch. 4
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p.49
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9620
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Empiricists base numbers on objects, Platonists base them on properties
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Ch. 4
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p.49
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9621
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Mathematics represents the world through structurally similar models.
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Ch. 4
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p.53
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9622
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'There are two apples' can be expressed logically, with no mention of numbers
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Ch. 4
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p.59
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9625
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To see a structure in something, we must already have the idea of the structure
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Ch. 4
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p.61
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9628
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Sets seem basic to mathematics, but they don't suit structuralism
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Ch. 5
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p.62
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9629
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For nomalists there are no numbers, only numerals
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Ch. 5
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p.63
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9630
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The most brilliant formalist was Hilbert
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Ch. 5
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p.65
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9634
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Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
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Ch. 5
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p.66
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9635
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Given atomism at one end, and a finite universe at the other, there are no physical infinities
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Ch. 5
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p.71
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9638
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Berry's Paradox finds a contradiction in the naming of huge numbers
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Ch. 6
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p.89
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9639
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Does some mathematics depend entirely on notation?
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Ch. 6
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p.92
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9640
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A term can have not only a sense and a reference, but also a 'computational role'
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Ch. 7
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p.94
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9641
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Definitions should be replaceable by primitives, and should not be creative
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Ch. 7
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p.97
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9642
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A flock of birds is not a set, because a set cannot go anywhere
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Ch. 7
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p.102
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9643
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Set theory may represent all of mathematics, without actually being mathematics
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Ch. 7
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p.105
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9644
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When graphs are defined set-theoretically, that won't cover unlabelled graphs
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Ch. 8
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p.113
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9645
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Constructivists say p has no value, if the value depends on Goldbach's Conjecture
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Ch. 9
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p.130
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9646
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There is no limit to how many ways something can be proved in mathematics
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Ch.10
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p.154
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9647
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Computers played an essential role in proving the four-colour theorem of maps
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Ch.10
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p.164
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9648
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π is a 'transcendental' number, because it is not the solution of an equation
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Ch.10
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p.170
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9649
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Axioms are either self-evident, or stipulations, or fallible attempts
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