Ideas of James Robert Brown, by Theme

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

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2. Reason / D. Definition / 2. Aims of Definition
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
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. Na´ve logical sets
Na´ve set theory assumed that there is a set for every condition
Nowadays conditions are only defined on existing sets
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The 'iterative' view says sets start with the empty set and build up
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
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
If a proposition is false, then its negation is true
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Axioms are either self-evident, or stipulations, or fallible attempts
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox finds a contradiction in the naming of huge numbers
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is the only place where we are sure we are right
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
'There are two apples' can be expressed logically, with no mention of numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / n. Pi
π is a 'transcendental' number, because it is not the solution of an equation
6. Mathematics / A. Nature of Mathematics / 5. Geometry
Bolzano wanted to reduce all of geometry to arithmetic
6. Mathematics / A. Nature of Mathematics / 6. Proof in Mathematics
There is no limit to how many ways something can be proved in mathematics
Computers played an essential role in proving the four-colour theorem of maps
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
Mathematics represents the world through structurally similar models.
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Set theory may represent all of mathematics, without actually being mathematics
When graphs are defined set-theoretically, that won't cover unlabelled graphs
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
To see a structure in something, we must already have the idea of the structure
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
Different versions of set theory result in different underlying structures for numbers
Sets seem basic to mathematics, but they don't suit structuralism
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
The irrationality of root-2 was achieved by intellect, not experience
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
There is an infinity of mathematical objects, so they can't be physical
Numbers are not abstracted from particulars, because each number is a particular
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Empiricists base numbers on objects, Platonists base them on properties
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Does some mathematics depend entirely on notation?
For nomalists there are no numbers, only numerals
The most brilliant formalist was Hilbert
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
There are no constructions for many highly desirable results in mathematics
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
David's 'Napoleon' is about something concrete and something abstract
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
The greatest discovery in human thought is Plato's discovery of abstract objects
18. Thought / E. Abstraction / 1. Abstract Thought
'Abstract' nowadays means outside space and time, not concrete, not physical
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
A term can have not only a sense and a reference, but also a 'computational role'
27. Natural Reality / D. Cosmology / 3. Infinite in Nature
Given atomism at one end, and a finite universe at the other, there are no physical infinities