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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

9604

Mathematics is the only place where we are sure we are right

Ch. 1

p.5

9605

If a proposition is false, then its negation is true

Ch. 1

p.5

9606

The irrationality of root2 was achieved by intellect, not experience

Ch. 2

p.8

9607

The greatest discovery in human thought is Plato's discovery of abstract objects

Ch. 2

p.12

9611

'Abstract' nowadays means outside space and time, not concrete, not physical

Ch. 2

p.12

9609

The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars

Ch. 2

p.12

9608

There are no constructions for many highly desirable results in mathematics

Ch. 2

p.12

9612

There is an infinity of mathematical objects, so they can't be physical

Ch. 2

p.12

9610

Numbers are not abstracted from particulars, because each number is a particular

Ch. 2

p.19

9615

Nowadays conditions are only defined on existing sets

Ch. 2

p.19

9613

Naïve set theory assumed that there is a set for every condition

Ch. 2

p.22

9617

The 'iterative' view says sets start with the empty set and build up

Ch. 3

p.28

9618

Bolzano wanted to reduce all of geometry to arithmetic

Ch. 3

p.40

9619

David's 'Napoleon' is about something concrete and something abstract

Ch. 4

p.49

9620

Empiricists base numbers on objects, Platonists base them on properties

Ch. 4

p.49

9621

Mathematics represents the world through structurally similar models.

Ch. 4

p.53

9622

'There are two apples' can be expressed logically, with no mention of numbers

Ch. 4

p.59

9625

To see a structure in something, we must already have the idea of the structure

Ch. 4

p.60

9627

Different versions of set theory result in different underlying structures for numbers

Ch. 4

p.61

9628

Sets seem basic to mathematics, but they don't suit structuralism

Ch. 5

p.62

9629

For nomalists there are no numbers, only numerals

Ch. 5

p.63

9630

The most brilliant formalist was Hilbert

Ch. 5

p.65

9634

Set theory says that natural numbers are an actual infinity (to accommodate their powerset)

Ch. 5

p.66

9635

Given atomism at one end, and a finite universe at the other, there are no physical infinities

Ch. 5

p.71

9638

Berry's Paradox finds a contradiction in the naming of huge numbers

Ch. 6

p.89

9639

Does some mathematics depend entirely on notation?

Ch. 6

p.92

9640

A term can have not only a sense and a reference, but also a 'computational role'

Ch. 7

p.94

9641

Definitions should be replaceable by primitives, and should not be creative

Ch. 7

p.97

9642

A flock of birds is not a set, because a set cannot go anywhere

Ch. 7

p.102

9643

Set theory may represent all of mathematics, without actually being mathematics

Ch. 7

p.105

9644

When graphs are defined settheoretically, that won't cover unlabelled graphs

Ch. 8

p.113

9645

Constructivists say p has no value, if the value depends on Goldbach's Conjecture

Ch. 9

p.130

9646

There is no limit to how many ways something can be proved in mathematics

Ch.10

p.154

9647

Computers played an essential role in proving the fourcolour theorem of maps

Ch.10

p.164

9648

π is a 'transcendental' number, because it is not the solution of an equation

Ch.10

p.170

9649

Axioms are either selfevident, or stipulations, or fallible attempts
