74 ideas
| 9641 | Definitions should be replaceable by primitives, and should not be creative [Brown,JR] |
| 12442 | 'Mickey Mouse is a fictional mouse' is true without a truthmaker [Azzouni] |
| 12439 | Truth is dispensable, by replacing truth claims with the sentence itself [Azzouni] |
| 12437 | Truth lets us assent to sentences we can't explicitly exhibit [Azzouni] |
| 9634 | Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR] |
| 10676 | The Axiom of Choice is a non-logical principle of set-theory [Hossack] |
| 10686 | The Axiom of Choice guarantees a one-one correspondence from sets to ordinals [Hossack] |
| 9615 | Nowadays conditions are only defined on existing sets [Brown,JR] |
| 9613 | Naïve set theory assumed that there is a set for every condition [Brown,JR] |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| 9617 | The 'iterative' view says sets start with the empty set and build up [Brown,JR] |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| 9642 | A flock of birds is not a set, because a set cannot go anywhere [Brown,JR] |
| 10687 | Maybe we reduce sets to ordinals, rather than the other way round [Hossack] |
| 10677 | Extensional mereology needs two definitions and two axioms [Hossack] |
| 9605 | If a proposition is false, then its negation is true [Brown,JR] |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| 12446 | Names function the same way, even if there is no object [Azzouni] |
| 10671 | Plural definite descriptions pick out the largest class of things that fit the description [Hossack] |
| 10666 | Plural reference will refer to complex facts without postulating complex things [Hossack] |
| 10669 | Plural reference is just an abbreviation when properties are distributive, but not otherwise [Hossack] |
| 10675 | A plural comprehension principle says there are some things one of which meets some condition [Hossack] |
| 9649 | Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR] |
| 9638 | Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR] |
| 10673 | Plural language can discuss without inconsistency things that are not members of themselves [Hossack] |
| 9604 | Mathematics is the only place where we are sure we are right [Brown,JR] |
| 9622 | 'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR] |
| 10680 | The theory of the transfinite needs the ordinal numbers [Hossack] |
| 10684 | I take the real numbers to be just lengths [Hossack] |
| 9648 | π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR] |
| 9621 | Mathematics represents the world through structurally similar models. [Brown,JR] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| 9646 | There is no limit to how many ways something can be proved in mathematics [Brown,JR] |
| 9647 | Computers played an essential role in proving the four-colour theorem of maps [Brown,JR] |
| 10674 | A plural language gives a single comprehensive induction axiom for arithmetic [Hossack] |
| 10681 | In arithmetic singularists need sets as the instantiator of numeric properties [Hossack] |
| 10685 | Set theory is the science of infinity [Hossack] |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| 9625 | To see a structure in something, we must already have the idea of the structure [Brown,JR] |
| 9628 | Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR] |
| 9606 | The irrationality of root-2 was achieved by intellect, not experience [Brown,JR] |
| 23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
| 9610 | Numbers are not abstracted from particulars, because each number is a particular [Brown,JR] |
| 9612 | There is an infinity of mathematical objects, so they can't be physical [Brown,JR] |
| 9620 | Empiricists base numbers on objects, Platonists base them on properties [Brown,JR] |
| 9630 | The most brilliant formalist was Hilbert [Brown,JR] |
| 9629 | For nomalists there are no numbers, only numerals [Brown,JR] |
| 9639 | Does some mathematics depend entirely on notation? [Brown,JR] |
| 9608 | There are no constructions for many highly desirable results in mathematics [Brown,JR] |
| 9645 | Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR] |
| 12447 | That all existents have causal powers is unknowable; the claim is simply an epistemic one [Azzouni] |
| 9619 | David's 'Napoleon' is about something concrete and something abstract [Brown,JR] |
| 12445 | If fictional objects really don't exist, then they aren't abstract objects [Azzouni] |
| 12449 | Modern metaphysics often derives ontology from the logical forms of sentences [Azzouni] |
| 10668 | We are committed to a 'group' of children, if they are sitting in a circle [Hossack] |
| 12440 | If objectual quantifiers ontologically commit, so does the metalanguage for its semantics [Azzouni] |
| 12438 | In the vernacular there is no unequivocal ontological commitment [Azzouni] |
| 12441 | We only get ontology from semantics if we have already smuggled it in [Azzouni] |
| 12448 | Things that don't exist don't have any properties [Azzouni] |
| 10664 | Complex particulars are either masses, or composites, or sets [Hossack] |
| 10678 | The relation of composition is indispensable to the part-whole relation for individuals [Hossack] |
| 10682 | The fusion of five rectangles can decompose into more than five parts that are rectangles [Hossack] |
| 10665 | Leibniz's Law argues against atomism - water is wet, unlike water molecules [Hossack] |
| 10663 | A thought can refer to many things, but only predicate a universal and affirm a state of affairs [Hossack] |
| 9611 | 'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR] |
| 9609 | The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR] |
| 9640 | A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR] |
| 9635 | Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR] |
| 10683 | We could ignore space, and just talk of the shape of matter [Hossack] |
| 12450 | The periodic table not only defines the elements, but also excludes other possible elements [Azzouni] |