14 ideas
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| 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 [Hossack] |
| 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 [Hossack] |
| 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. |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
| 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 [Hossack] |
| 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. |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| 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. |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| 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 [Hossack] |
| 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. |
| 19743 | A notebook counts as memory, if is available to consciousness and guides our actions [Clark/Chalmers] |
| Full Idea: Beliefs are partly constituted by features of the environment. ....a notebook plays for one person the same role that memory plays for another. ...The information is reliably there, available to consciousness, and to guide action, just as belief is. | |
| From: A Clark / D Chalmers (The Extended Mind [1998], §4) | |
| A reaction: This is the modern externalist approach to beliefs (along with broad content and external cognition systems). Not quite what we used to mean by beliefs, but we'll get used to it. I believe Plato wrote what it said in his books. Is memory just a role? |
| 6176 | A mechanism can count as 'cognitive' whether it is in the brain or outside it [Clark/Chalmers, by Rowlands] |
| Full Idea: If the operation of a brain implant inside the brain is a cognitive operation, why should it not count as a cognitive operation when it is outside the brain? There are many mechanisms which would count as cognitive if they were inside the subject. | |
| From: report of A Clark / D Chalmers (The Extended Mind [1998]) by Mark Rowlands - Externalism Ch.9 | |
| A reaction: This argues for externalism of the vehicle of thought, rather than its content. The idea is that there is no significant difference between remembering a phone number and writing it on a bit of paper. I find it hard to disagree. |
| 19741 | If something in the world could equally have been a mental process, it is part of our cognition [Clark/Chalmers] |
| Full Idea: If, as we confront some task, a part of the world functions as a process which, were it done in the head, we would have no hesitation in recognising as part of the cognitive process, then that part of the world is part of the cognitive process. | |
| From: A Clark / D Chalmers (The Extended Mind [1998], §2) | |
| A reaction: In some sense they are obviously right that our cognitive activities spill out into books, calculators, record-keeping. It seems more like an invitation to shift the meaning of the word 'mind', than a proof that we have got it wrong. |
| 19742 | Consciousness may not extend beyond the head, but cognition need not be conscious [Clark/Chalmers] |
| Full Idea: Many identify the cognitive with the conscious, and it seems far from plausible that consciousness extends outside the head in these cases. But not every cognitive process, at least on standard usage, is a conscious process. | |
| From: A Clark / D Chalmers (The Extended Mind [1998], §3) | |
| A reaction: This gives you two sorts of externalism about mind to consider. No, three, if you say there is extended conceptual content, then extended cognition processes, then extended consciousness. Depends what you mean by 'consciousness'. |
| 19744 | If a person relies on their notes, those notes are parted of the extended system which is the person [Clark/Chalmers] |
| Full Idea: If Otto relies on his notebook, what this comes to is that Otto himself is best regarded as an extended system, a coupling of biological organism and external resources. | |
| From: A Clark / D Chalmers (The Extended Mind [1998], §5) | |
| A reaction: You start to get giddy as you read this stuff. If two people constantly share a notebook, they begin to blend into one another. It inclines me towards a more 'animalist' view of the nature of a person or a self. |
| 6017 | Nomos is king [Pindar] |
| Full Idea: Nomos is king. | |
| From: Pindar (poems [c.478 BCE], S 169), quoted by Thomas Nagel - The Philosophical Culture | |
| A reaction: This seems to be the earliest recorded shot in the nomos-physis wars (the debate among sophists about moral relativism). It sounds as if it carries the full relativist burden - that all that matters is what has been locally decreed. |