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 | 9606 | The irrationality of root-2 was achieved by intellect, not experience |
Ch. 1 | p.5 | 9605 | If a proposition is false, then its negation is true |
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 | 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.12 | 9608 | There are no constructions for many highly desirable results in mathematics |
Ch. 2 | p.19 | 9613 | Naïve set theory assumed that there is a set for every condition |
Ch. 2 | p.19 | 9615 | Nowadays conditions are only defined on existing sets |
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 set-theoretically, 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 four-colour 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 self-evident, or stipulations, or fallible attempts |