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

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

9613	Naïve set theory assumed that there is a set for every condition

9615	Nowadays conditions are only defined on existing sets

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

9630	The most brilliant formalist was Hilbert

9639	Does some mathematics depend entirely on notation?

9629	For nomalists there are no numbers, only numerals

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

9645	Constructivists say p has no value, if the value depends on Goldbach's Conjecture

9608	There are no constructions for many highly desirable results in mathematics

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