### Ideas from 'Philosophy of Mathematics' by James Robert Brown , by Theme Structure

#### [found in 'Philosophy of Mathematics' by Brown,James Robert [Routledge 2002,0-415-12275-9]].

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###### 2. Reason / D. Definition / 2. Aims of Definition
 9641 Definitions should be replaceable by primitives, and should not be creative
###### 4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
 9634 Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
 9615 Nowadays conditions are only defined on existing sets
 9613 Naïve set theory assumed that there is a set for every condition
###### 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
 9617 The 'iterative' view says sets start with the empty set and build up
###### 4. Formal Logic / F. Set Theory ST / 7. Natural Sets
 9642 A flock of birds is not a set, because a set cannot go anywhere
###### 5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
 9605 If a proposition is false, then its negation is true
###### 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
 9649 Axioms are either self-evident, or stipulations, or fallible attempts
###### 5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
 9638 Berry's Paradox finds a contradiction in the naming of huge numbers
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 9604 Mathematics is the only place where we are sure we are right
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
 9622 'There are two apples' can be expressed logically, with no mention of numbers
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / n. Pi
 9648 π is a 'transcendental' number, because it is not the solution of an equation
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 9621 Mathematics represents the world through structurally similar models.
###### 6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
 9646 There is no limit to how many ways something can be proved in mathematics
 9647 Computers played an essential role in proving the four-colour theorem of maps
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
 9643 Set theory may represent all of mathematics, without actually being mathematics
 9644 When graphs are defined set-theoretically, that won't cover unlabelled graphs
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
 9625 To see a structure in something, we must already have the idea of the structure
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
 9628 Sets seem basic to mathematics, but they don't suit structuralism
###### 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
###### 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
 9610 Numbers are not abstracted from particulars, because each number is a particular
###### 6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 9620 Empiricists base numbers on objects, Platonists base them on properties
###### 6. Mathematics / C. Sources of Mathematics / 7. Formalism
 9639 Does some mathematics depend entirely on notation?
 9629 For nomalists there are no numbers, only numerals
 9630 The most brilliant formalist was Hilbert
###### 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
 9608 There are no constructions for many highly desirable results in mathematics
 9645 Constructivists say p has no value, if the value depends on Goldbach's Conjecture
###### 7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
 9619 David's 'Napoleon' is about something concrete and something abstract
###### 18. Thought / E. Abstraction / 1. Abstract Thought
 9611 'Abstract' nowadays means outside space and time, not concrete, not physical
 9609 The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
###### 19. Language / A. Nature of Meaning / 7. Meaning Holism / c. Meaning by Role
 9640 A term can have not only a sense and a reference, but also a 'computational role'
###### 26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
 9635 Given atomism at one end, and a finite universe at the other, there are no physical infinities