10676 | The Axiom of Choice is a non-logical principle of set-theory |
Full Idea: The Axiom of Choice seems better treated as a non-logical principle of set-theory. | |
From: Keith Hossack (Plurals and Complexes [2000], 4 n8) | |
A reaction: This reinforces the idea that set theory is not part of logic (and so pure logicism had better not depend on set theory). |
10686 | The Axiom of Choice guarantees a one-one correspondence from sets to ordinals |
Full Idea: We cannot explicitly define one-one correspondence from the sets to the ordinals (because there is no explicit well-ordering of R). Nevertheless, the Axiom of Choice guarantees that a one-one correspondence does exist, even if we cannot define it. | |
From: Keith Hossack (Plurals and Complexes [2000], 10) |
23623 | Predicativism says only predicated sets exist |
Full Idea: Predicativists doubt the existence of sets with no predicative definition. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 02.3) | |
A reaction: This would imply that sets which encounter paradoxes when they try to be predicative do not therefore exist. Surely you can have a set of random objects which don't fall under a single predicate? |
23624 | The iterative conception has to appropriate Replacement, to justify the ordinals |
Full Idea: The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
A reaction: The modern approach to axioms, where we want to prove something so we just add an axiom that does the job. |
23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set |
Full Idea: The limitation of size conception of sets justifies the axiom of Replacement, but cannot justify Power Set, so NBG set theory appropriates the Power Set axiom from ZFC. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
A reaction: Which suggests that the Power Set axiom is not as indispensable as it at first appears to be. |
10687 | Maybe we reduce sets to ordinals, rather than the other way round |
Full Idea: We might reduce sets to ordinal numbers, thereby reversing the standard set-theoretical reduction of ordinals to sets. | |
From: Keith Hossack (Plurals and Complexes [2000], 10) | |
A reaction: He has demonstrated that there are as many ordinals as there are sets. |
10677 | Extensional mereology needs two definitions and two axioms |
Full Idea: Extensional mereology defs: 'distinct' things have no parts in common; a 'fusion' has some things all of which are parts, with no further parts. Axioms: (transitivity) a part of a part is part of the whole; (sums) any things have a unique fusion. | |
From: Keith Hossack (Plurals and Complexes [2000], 5) |
23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' |
Full Idea: The sentence connective 'and' also has an order-sensitive meaning, when it means something like 'and then'. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.4) | |
A reaction: This is support the idea that orders are a feature of reality, just as much as possible concatenation. Relational predicates, he says, refer to series rather than to individuals. Nice point. |
23627 | 'Before' and 'after' are not two relations, but one relation with two orders |
Full Idea: The reason the two predicates 'before' and 'after' are needed is not to express different relations, but to indicate its order. Since there can be difference of order without difference of relation, the nature of relations is not the source of order. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.3) | |
A reaction: This point is to refute Russell's 1903 claim that order arises from the nature of relations. Hossack claims that it is ordered series which are basic. I'm inclined to agree with him. |
10671 | Plural definite descriptions pick out the largest class of things that fit the description |
Full Idea: If we extend the power of language with plural definite descriptions, these would pick out the largest class of things that fit the description. | |
From: Keith Hossack (Plurals and Complexes [2000], 3) |
10666 | Plural reference will refer to complex facts without postulating complex things |
Full Idea: It may be that plural reference gives atomism the resources to state complex facts without needing to refer to complex things. | |
From: Keith Hossack (Plurals and Complexes [2000], 1) | |
A reaction: This seems the most interesting metaphysical implication of the possibility of plural quantification. |
10669 | Plural reference is just an abbreviation when properties are distributive, but not otherwise |
Full Idea: If all properties are distributive, plural reference is just a handy abbreviation to avoid repetition (as in 'A and B are hungry', to avoid 'A is hungry and B is hungry'), but not all properties are distributive (as in 'some people surround a table'). | |
From: Keith Hossack (Plurals and Complexes [2000], 2) | |
A reaction: The characteristic examples to support plural quantification involve collective activity and relations, which might be weeded out of our basic ontology, thus leaving singular quantification as sufficient. |
10675 | A plural comprehension principle says there are some things one of which meets some condition |
Full Idea: Singular comprehension principles have a bad reputation, but the plural comprehension principle says that given a condition on individuals, there are some things such that something is one of them iff it meets the condition. | |
From: Keith Hossack (Plurals and Complexes [2000], 4) |
10673 | Plural language can discuss without inconsistency things that are not members of themselves |
Full Idea: In a plural language we can discuss without fear of inconsistency the things that are not members of themselves. | |
From: Keith Hossack (Plurals and Complexes [2000], 4) | |
A reaction: [see Hossack for details] |
10680 | The theory of the transfinite needs the ordinal numbers |
Full Idea: The theory of the transfinite needs the ordinal numbers. | |
From: Keith Hossack (Plurals and Complexes [2000], 8) |
10684 | I take the real numbers to be just lengths |
Full Idea: I take the real numbers to be just lengths. | |
From: Keith Hossack (Plurals and Complexes [2000], 9) | |
A reaction: I love it. Real numbers are beginning to get on my nerves. They turn up to the party with no invitation and improperly dressed, and then refuse to give their names when challenged. |
23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability |
Full Idea: The transfinite ordinal numbers are important in the theory of proofs, and essential in the theory of recursive functions and computability. Mathematics would be incomplete without them. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 10.1) | |
A reaction: Hossack offers this as proof that the numbers are not human conceptual creations, but must exist beyond the range of our intellects. Hm. |
10674 | A plural language gives a single comprehensive induction axiom for arithmetic |
Full Idea: A language with plurals is better for arithmetic. Instead of a first-order fragment expressible by an induction schema, we have the complete truth with a plural induction axiom, beginning 'If there are some numbers...'. | |
From: Keith Hossack (Plurals and Complexes [2000], 4) |
10681 | In arithmetic singularists need sets as the instantiator of numeric properties |
Full Idea: In arithmetic singularists need sets as the instantiator of numeric properties. | |
From: Keith Hossack (Plurals and Complexes [2000], 8) |
10685 | Set theory is the science of infinity |
Full Idea: Set theory is the science of infinity. | |
From: Keith Hossack (Plurals and Complexes [2000], 10) |
23621 | Numbers are properties, not sets (because numbers are magnitudes) |
Full Idea: I propose that numbers are properties, not sets. Magnitudes are a kind of property, and numbers are magnitudes. …Natural numbers are properties of pluralities, positive reals of continua, and ordinals of series. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro) | |
A reaction: Interesting! Since time can have a magnitude (three weeks) just as liquids can (three litres), it is not clear that there is a single natural property we can label 'magnitude'. Anything we can manage to measure has a magnitude. |
23622 | We can only mentally construct potential infinities, but maths needs actual infinities |
Full Idea: Numbers cannot be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers. | |
From: Keith Hossack (Knowledge and the Philosophy of Number [2020], Intro 2) | |
A reaction: Doubt this, but don't know enough to refute it. Actual infinities were a fairly late addition to maths, I think. I would think treating fictional complete infinities as real would be sufficient for the job. Like journeys which include imagined roads. |
10668 | We are committed to a 'group' of children, if they are sitting in a circle |
Full Idea: By Quine's test of ontological commitment, if some children are sitting in a circle, no individual child can sit in a circle, so a singular paraphrase will have us committed to a 'group' of children. | |
From: Keith Hossack (Plurals and Complexes [2000], 2) | |
A reaction: Nice of why Quine is committed to the existence of sets. Hossack offers plural quantification as a way of avoiding commitment to sets. But is 'sitting in a circle' a real property (in the Shoemaker sense)? I can sit in a circle without realising it. |
10664 | Complex particulars are either masses, or composites, or sets |
Full Idea: Complex particulars are of at least three types: masses (which sum, of which we do not ask 'how many?' but 'how much?'); composite individuals (how many?, and summing usually fails); and sets (only divisible one way, unlike composites). | |
From: Keith Hossack (Plurals and Complexes [2000], 1) | |
A reaction: A composite pile of grains of sand gradually becomes a mass, and drops of water become 'water everywhere'. A set of people divides into individual humans, but redescribe the elements as the union of males and females? |
10678 | The relation of composition is indispensable to the part-whole relation for individuals |
Full Idea: The relation of composition seems to be indispensable in a correct account of the part-whole relation for individuals. | |
From: Keith Hossack (Plurals and Complexes [2000], 7) | |
A reaction: This is the culmination of a critical discussion of mereology and ontological atomism. At first blush it doesn't look as if 'composition' has much chance of being a precise notion, and it will be plagued with vagueness. |
10665 | Leibniz's Law argues against atomism - water is wet, unlike water molecules |
Full Idea: We can employ Leibniz's Law against mereological atomism. Water is wet, but no water molecule is wet. The set of infinite numbers is infinite, but no finite number is infinite. ..But with plural reference the atomist can resist this argument. | |
From: Keith Hossack (Plurals and Complexes [2000], 1) | |
A reaction: The idea of plural reference is to state plural facts without referring to complex things, which is interesting. The general idea is that we have atomism, and then all the relations, unities, identities etc. are in the facts, not in the things. I like it. |
10682 | The fusion of five rectangles can decompose into more than five parts that are rectangles |
Full Idea: The fusion of five rectangles may have a decomposition into more than five parts that are rectangles. | |
From: Keith Hossack (Plurals and Complexes [2000], 8) |
10663 | A thought can refer to many things, but only predicate a universal and affirm a state of affairs |
Full Idea: A thought can refer to a particular or a universal or a state of affairs, but it can predicate only a universal and it can affirm only a state of affairs. | |
From: Keith Hossack (Plurals and Complexes [2000], 1) | |
A reaction: Hossack is summarising Armstrong's view, which he is accepting. To me, 'thought' must allow for animals, unlike language. I think Hossack's picture is much too clear-cut. Do animals grasp universals? Doubtful. Can they predicate? Yes. |
10683 | We could ignore space, and just talk of the shape of matter |
Full Idea: We might dispense with substantival space, and say that if the distribution of matter in space could have been different, that just means the matter of the Universe could have been shaped differently (with geometry as the science of shapes). | |
From: Keith Hossack (Plurals and Complexes [2000], 9) |