16 ideas
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
| A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives'). | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology. |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro) | |
| A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it. |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1) | |
| A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application. |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |
| Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers. | |
| From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2) | |
| A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated. |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 9312 | Consciousness is reductively explained either by how it represents, or how it is represented [Kriegel/Williford] |
| Full Idea: The two main competitors for reductive theories of consciousness are the representational theory (conscious if it represents in the right way), and higher-order monitoring (conscious if it is represented in the right way). | |
| From: U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], Intro) | |
| A reaction: Presumably there are also neuroscientists hunting for physical functions which might generate consciousness. The two mentioned here are rivals at one level of discourse. Both views may be simplistic, if complex teams of activities are involved. |
| 9313 | Experiences can be represented consciously or unconsciously, so representation won't explain consciousness [Kriegel/Williford] |
| Full Idea: On the assumption that any environmental feature can be represented either consciously or unconsciously, it is unclear how the mere representation of such a feature can render the representing state conscious. | |
| From: U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §1) | |
| A reaction: The authors are rejecting simple representation as the key, in favour of a distinctive sort of self-representation. I'm inclined to think that consciousness results from multiple co-ordinated layers of representation etc., which has no simple account. |
| 9315 | Red tomato experiences are conscious if the state represents the tomato and itself [Kriegel/Williford] |
| Full Idea: The self-representational theory of consciousness says that when one has a conscious experience as of a red tomato, one is in an internal state that represents both a red tomato and itself. | |
| From: U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §1) | |
| A reaction: This seems to be avoiding the concept of 'higher-order', and yet that seems the only way to describe it - thought steps outside of itself, generating a level of meta-thought. I think that's the way to go. Philosophy is about-fifth level. |
| 9316 | How is self-representation possible, does it produce a regress, and is experience like that? [Kriegel/Williford] |
| Full Idea: The difficulties with a self-representational view of consciousness are how self-representation of mental states could be possible, whether it leads to an infinite regress, and whether it can capture the actual phenomenology of experience. | |
| From: U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §3) | |
| A reaction: [compressed] All of these objections strike me as persuasive, especially the first one. I'm not sure I know what self-representation is. Mirrors externally represent, and they can't represent themselves. Two mirrors together achieve something.. |
| 9314 | Unfortunately, higher-order representations could involve error [Kriegel/Williford] |
| Full Idea: A problem for explaining consciousness by higher-order representations is that, like their first-order counterparts, they can misrepresent; there could be a subjective impression of being in a conscious state without actually being in any conscious state. | |
| From: U Kriegel / K Williford (Intro to 'Self-Representational Consciousness' [2006], §1) | |
| A reaction: It sounds plausible that this is a logical possibility, but how do you assess whether it is an actual or natural possibility? Are we saying that higher-order representations are judgments, which could be true or false? Hm. |