24 ideas
| 9921 | 'True' is only occasionally useful, as in 'everything Fermat believed was true' [Burgess/Rosen] |
| Full Idea: In the disquotational view of truth, what saves truth from being wholly redundant and so wholly useless, is mainly that it provides an ability to state generalisations like 'Everything Fermat believed was true'. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], I.A.2.c) | |
| A reaction: Sounds like the thin end of the wedge. Presumably we can infer that the first thing Fermat believed on his last Christmas Day was true. |
| 9924 | Modal logic gives an account of metalogical possibility, not metaphysical possibility [Burgess/Rosen] |
| Full Idea: If you want a logic of metaphysical possibility, the existing literature was originally developed to supply a logic of metalogical possibility, and still reflects its origins. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.B.3.b) | |
| A reaction: This is a warning shot (which I don't fully understand) to people like me, who were beginning to think they could fill their ontology with possibilia, which could then be incorporated into the wider account of logical thinking. Ah well... |
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 9933 | The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set [Burgess/Rosen] |
| Full Idea: The paradoxes only seem to arise in connection with Frege's logical notion of extension or class, not Cantor's mathematical notion of set. Cantor never assumed that every condition determines a set. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], III.C.1.b) | |
| A reaction: This makes the whole issue a parochial episode in the history of philosophy, not a central question. Cantor favoured some sort of abstractionism (see Kit Fine on the subject). |
| 9928 | Mereology implies that acceptance of entities entails acceptance of conglomerates [Burgess/Rosen] |
| Full Idea: Mereology has ontological implications. The acceptance of some initial entities involves the acceptance of many further entities, arbitrary wholes having the entities as parts. It must accept conglomerates. Geometric points imply geometric regions. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.1.b) | |
| A reaction: Presumably without the wholes being entailed by the parts, there is no subject called 'mereology'. But if the conglomeration is unrestricted, there is not much left to be said. 'Restricted' composition (by nature?) sounds a nice line. |
| 9926 | A relation is either a set of sets of sets, or a set of sets [Burgess/Rosen] |
| Full Idea: While in general a relation is taken to be a set of ordered pairs <u, v> = {{u}, {u, v}}, and hence a set of sets of sets, in special cases a relation can be represented by a set of sets. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.1.a) | |
| A reaction: [See book for their examples, which are <, symmetric, and arbitrary] The fact that a relation (or anything else) can be represented in a certain way should never ever be taken to mean that you now know what the thing IS. |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 9932 | The paradoxes no longer seem crucial in critiques of set theory [Burgess/Rosen] |
| Full Idea: Recent commentators have de-emphasised the set paradoxes because they play no prominent part in motivating the most articulate and active opponents of set theory, such as Kronecker (constructivism) or Brouwer (intuitionism), or Weyl (predicativism). | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], III.C.1.b) | |
| A reaction: This seems to be a sad illustration of the way most analytical philosophers have to limp along behind the logicians and mathematicians, arguing furiously about problems that have largely been abandoned. |
| 9923 | We should talk about possible existence, rather than actual existence, of numbers [Burgess/Rosen] |
| Full Idea: The modal strategy for numbers is to replace assumptions about the actual existence of numbers by assumptions about the possible existence of numbers | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.B.3.a) | |
| A reaction: This seems to be quite a good way of dealing with very large numbers and infinities. It is not clear whether 5 is so regularly actualised that we must consider it as permanent, or whether it is just a prominent permanent possibility. |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 9925 | Structuralism and nominalism are normally rivals, but might work together [Burgess/Rosen] |
| Full Idea: Usually structuralism and nominalism are considered rivals. But structuralism can also be the first step in a strategy of nominalist reconstrual or paraphrase. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.0) | |
| A reaction: Hellman and later Chihara seem to be the main proponents of nominalist structuralism. My sympathies lie with this strategy. Are there objects at the nodes of the structure, or is the structure itself platonic? Mill offers a route. |
| 9934 | Number words became nouns around the time of Plato [Burgess/Rosen] |
| Full Idea: The transition from using number words purely as adjectives to using them extensively as nouns has been traced to 'around the time of Plato'. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], III.C.2.a) | |
| A reaction: [The cite Kneale and Kneale VI,§2 for this] It is just too tempting to think that in fact Plato (and early Platonists) were totally responsible for this shift, since the whole reification of numbers seems to be inherently platonist. |
| 9918 | Abstract/concrete is a distinction of kind, not degree [Burgess/Rosen] |
| Full Idea: The distinction of abstract and concrete is one of kind and not degree. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], I.A.1.a) | |
| A reaction: I think I must agree with this. If there is a borderline, it would be in particulars that seem to have an abstract aspect to them. A horse involves the abstraction of being a horse, and it involves be one horse. |
| 9929 | Much of what science says about concrete entities is 'abstraction-laden' [Burgess/Rosen] |
| Full Idea: Much of what science says about concrete entities is 'abstraction-laden'. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], III.A.1.d) | |
| A reaction: Not just science. In ordinary conversation we continually refer to particulars using so-called 'universal' predicates and object-terms, which are presumably abstractions. 'I've just seen an elephant'. |
| 9927 | Mathematics has ascended to higher and higher levels of abstraction [Burgess/Rosen] |
| Full Idea: In mathematics, since the beginning of the nineteenth century, there has been an ascent to higher and higher levels of abstraction. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.C.1.b) | |
| A reaction: I am interested in clarifying what this means, which might involve the common sense and psychological view of the matter, as well as some sort of formal definition in terms of equivalence (or whatever). |
| 9930 | Abstraction is on a scale, of sets, to attributes, to type-formulas, to token-formulas [Burgess/Rosen] |
| Full Idea: There is a scale of abstractness that leads downwards from sets through attributes to formulas as abstract types and on to formulas as abstract tokens. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], III.B.2.c) | |
| A reaction: Presumably the 'abstract tokens' at the bottom must have some interpretation, to support the system. Presumably one can keep going upwards, through sets of sets of sets. |
| 9089 | Knowledge is a quality existing subjectively in the soul [William of Ockham] |
| Full Idea: Knowledge is a certain quality which exists in the soul as its subject ('existens subiective in anima'). | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: One might say here that knowledge is a property, and so it might not be susceptible to further analysis. It invites the question of how you could know by introspection that you have got it, which would be an extreme internalist view. |
| 9091 | Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham] |
| Full Idea: Sometimes 'knowledge' means evident cognition of the conclusion alone, sometimes of the demonstration as a whole. | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: 'Demonstration' will be something like Greek 'logos' - full understanding, ability to explain and give reasons. William is certainly right about normal usage. I know the answer in a quiz, without any requirement for justifications. |
| 9090 | Knowledge is certain cognition of something that is true [William of Ockham] |
| Full Idea: Knowledge is certain cognition of something that is true. | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: This view has problems. William is not facing up to the sceptical questions which can shake any degree of certainty, and also that someone who lacked self-confidence might know many things while always feeling uncertain about them. 'Cognition' must go! |
| 9919 | The old debate classified representations as abstract, not entities [Burgess/Rosen] |
| Full Idea: The original debate was over abstract ideas; thus it was mental (or linguistic) representations that were classified as abstract or otherwise, and not the entities represented. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], I.A.1.b) | |
| A reaction: This seems to beg the question of whether there are any such entities. It is equally plausible to talk of the entities that are 'constructed', rather than 'represented'. |
| 9922 | If space is really just a force-field, then it is a physical entity [Burgess/Rosen] |
| Full Idea: According to many philosophical commentators, a force-field must be considered to be a physical entity, and as the distinction between space and the force-field may be considered to be merely verbal, space itself may be considered to be a physical entity. | |
| From: JP Burgess / G Rosen (A Subject with No Object [1997], II.A.1) | |
| A reaction: The ontology becomes a bit odd if we cheerfully accept that space is physical, but then we can't give the same account of time. I'm not sure how time could be physical. What's it made of? |