21 ideas
| 9978 | Analytic philosophy focuses too much on forms of expression, instead of what is actually said [Tait] |
| Full Idea: The tendency to attack forms of expression rather than attempting to appreciate what is actually being said is one of the more unfortunate habits that analytic philosophy inherited from Frege. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IV) | |
| A reaction: The key to this, I say, is to acknowledge the existence of propositions (in brains). For example, this belief will make teachers more sympathetic to pupils who are struggling to express an idea, and verbal nit-picking becomes totally irrelevant. |
| 9986 | The null set was doubted, because numbering seemed to require 'units' [Tait] |
| Full Idea: The conception that what can be numbered is some object (including flocks of sheep) relative to a partition - a choice of unit - survived even in the late nineteenth century in the form of the rejection of the null set (and difficulties with unit sets). | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], IX) | |
| A reaction: This old view can't be entirely wrong! Frege makes the point that if asked to count a pack of cards, you must decide whether to count cards, or suits, or pips. You may not need a 'unit', but you need a concept. 'Units' name concept-extensions nicely! |
| 10653 | Maybe set theory need not be well-founded [Varzi] |
| Full Idea: There are some proposals for non-well-founded set theory (tolerating cases of self-membership and membership circularities). | |
| From: Achille Varzi (Mereology [2003], 2.1) | |
| A reaction: [He cites Aczel 1988, and Barwise and Moss 1996] |
| 9984 | We can have a series with identical members [Tait] |
| Full Idea: Why can't we have a series (as opposed to a linearly ordered set) all of whose members are identical, such as (a, a, a...,a)? | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VII) | |
| A reaction: The question is whether the items order themselves, which presumably the natural numbers are supposed to do, or whether we impose the order (and length) of the series. What decides how many a's there are? Do we order, or does nature? |
| 10648 | Mereology need not be nominalist, though it is often taken to be so [Varzi] |
| Full Idea: While mereology was originally offered with a nominalist viewpoint, resulting in a conception of mereology as an ontologically parsimonious alternative to set theory, there is no necessary link between analysis of parthood and nominalism. | |
| From: Achille Varzi (Mereology [2003], 1) | |
| A reaction: He cites Lesniewski and Leonard-and-Goodman. Do you allow something called a 'whole' into your ontology, as well as the parts? He observes that while 'wholes' can be concrete, they can also be abstract, if the parts are abstract. |
| 10655 | Are there mereological atoms, and are all objects made of them? [Varzi] |
| Full Idea: It is an open question whether there are any mereological atoms (with no proper parts), and also whether every object is ultimately made up of atoms. | |
| From: Achille Varzi (Mereology [2003], 3) | |
| A reaction: Such a view would have to presuppose (metaphysically) that the divisibility of matter has limits. If one follows this route, then are there only 'natural' wholes, or are we 'unrestricted' in our view of how the atoms combine? I favour the natural route. |
| 10659 | There is something of which everything is part, but no null-thing which is part of everything [Varzi] |
| Full Idea: It is common in mereology to hold that there is something of which everything is part, but few hold that there is a 'null entity' that is part of everything. | |
| From: Achille Varzi (Mereology [2003], 4.1) | |
| A reaction: This comes out as roughly the opposite of set theory, which cannot do without the null set, but is not keen on the set of everything. |
| 13416 | Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C] |
| Full Idea: The axiomatic conception of mathematics is the only viable one. ...But they are true because they are axioms, in contrast to the view advanced by Frege (to Hilbert) that to be a candidate for axiomhood a statement must be true. | |
| From: report of William W. Tait (Intro to 'Provenance of Pure Reason' [2005], p.4) by Charles Parsons - Review of Tait 'Provenance of Pure Reason' §2 | |
| A reaction: This looks like the classic twentieth century shift in the attitude to axioms. The Greek idea is that they must be self-evident truths, but the Tait-style view is that they are just the first steps in establishing a logical structure. I prefer the Greeks. |
| 10661 | 'Composition is identity' says multitudes are the reality, loosely composing single things [Varzi] |
| Full Idea: The thesis known as 'composition is identity' is that identity is mereological composition; a fusion is just the parts counted loosely, but it is strictly a multitude and loosely a single thing. | |
| From: Achille Varzi (Mereology [2003], 4.3) | |
| A reaction: [He cites D.Baxter 1988, in Mind] It is not clear, from this simple statement, what the difference is between multitudes that are parts of a thing, and multitudes that are not. A heavy weight seems to hang on the notion of 'composed of'. |
| 10647 | Parts may or may not be attached, demarcated, arbitrary, material, extended, spatial or temporal [Varzi] |
| Full Idea: The word 'part' can used whether it is attached, or arbitrarily demarcated, or gerrymandered, or immaterial, or unextended, or spatial, or temporal. | |
| From: Achille Varzi (Mereology [2003], 1) |
| 10651 | If 'part' is reflexive, then identity is a limit case of parthood [Varzi] |
| Full Idea: Taking reflexivity as constitutive of the meaning of 'part' amounts to regarding identity as a limit case of parthood. | |
| From: Achille Varzi (Mereology [2003], 2.1) | |
| A reaction: A nice thought, but it is horribly 'philosophical', and a long way from ordinary usage and common sense (which is, I'm sorry to say, a BAD thing). |
| 10649 | 'Part' stands for a reflexive, antisymmetric and transitive relation [Varzi] |
| Full Idea: It seems obvious that 'part' stands for a partial ordering, a reflexive ('everything is part of itself'), antisymmetic ('two things cannot be part of each other'), and transitive (a part of a part of a thing is part of that thing) relation. | |
| From: Achille Varzi (Mereology [2003], 2.1) | |
| A reaction: I'm never clear why the reflexive bit of the relation should be taken as 'obvious', since it seems to defy normal usage and common sense. It would be absurd to say 'I'll give you part of the cake' and hand you the whole of it. See Idea 10651. |
| 10654 | The parthood relation will help to define at least seven basic predicates [Varzi] |
| Full Idea: With a basic parthood relation, we can formally define various mereological predicates, such as overlap, underlap, proper part, over-crossing, under-crossing, proper overlap, and proper underlap. | |
| From: Achille Varzi (Mereology [2003], 2.2) | |
| A reaction: [Varzi offers some diagrams, but they need interpretation] |
| 10658 | Sameness of parts won't guarantee identity if their arrangement matters [Varzi] |
| Full Idea: We might say that sameness of parts is not sufficient for identity, as some entities may differ exclusively with respect to the arrangement of the parts, as when we compare 'John loves Mary' with 'Mary loves John'. | |
| From: Achille Varzi (Mereology [2003], 3.2) | |
| A reaction: Presumably wide dispersal should also prevent parts from fixing wholes, but there is so much vagueness here that it is tempting to go for unrestricted composition, and then work back to the common sense position. |
| 10652 | Conceivability may indicate possibility, but literary fantasy does not [Varzi] |
| Full Idea: Conceivability may well be a guide to possibility, but literary fantasy is by itself no evidence of conceivability. | |
| From: Achille Varzi (Mereology [2003], 2.1) | |
| A reaction: Very nice. People who cite 'conceivability' in this context often have a disgracefully loose usage for the word. Really, really conceivable is probably our only guide to possibility. |
| 16721 | Changes in secondary qualities are caused by changes in primary qualities [Giles of Orleans] |
| Full Idea: Every alteration in secondary qualities is caused by some alteration made in the primary qualities. | |
| From: Giles of Orleans (On 'Generation and Corruption' [1270], I.6c) | |
| A reaction: Pasnau calls this the 'supervenience' thesis, here with the addition of causation. Was this doctrine retained by Locke and Boyle? |
| 9981 | Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete [Tait] |
| Full Idea: If the sense of a proposition about the abstract domain is given in terms of the corresponding proposition about the (relatively) concrete domain, ..and the truth of the former is founded upon the truth of the latter, then this is 'logical abstraction'. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: The 'relatively' in parentheses allows us to apply his idea to levels of abstraction, and not just to the simple jump up from the concrete. I think Tait's proposal is excellent, rather than purloining 'abstraction' for an internal concept within logic. |
| 9982 | Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait] |
| Full Idea: Although (in Cantor and Dedekind) abstraction does not (as has often been observed) play any role in their proofs, but it does play a role, in that it fixes the grammar, the domain of meaningful propositions, and so determining the objects in the proofs. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: [compressed] This is part of a defence of abstractionism in Cantor and Dedekind (see K.Fine also on the subject). To know the members of a set, or size of a domain, you need to know the process or function which created the set. |
| 9985 | Abstraction may concern the individuation of the set itself, not its elements [Tait] |
| Full Idea: A different reading of abstraction is that it concerns, not the individuating properties of the elements relative to one another, but rather the individuating properties of the set itself, for example the concept of what is its extension. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], VIII) | |
| A reaction: If the set was 'objects in the room next door', we would not be able to abstract from the objects, but we might get to the idea of things being contain in things, or the concept of an object, or a room. Wrong. That's because they are objects... Hm. |
| 9972 | Why should abstraction from two equipollent sets lead to the same set of 'pure units'? [Tait] |
| Full Idea: Why should abstraction from two equipollent sets lead to the same set of 'pure units'? | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996]) | |
| A reaction: [Tait is criticising Cantor] This expresses rather better than Frege or Dummett the central problem with the abstractionist view of how numbers are derived from matching groups of objects. |
| 9980 | If abstraction produces power sets, their identity should imply identity of the originals [Tait] |
| Full Idea: If the power |A| is obtained by abstraction from set A, then if A is equipollent to set B, then |A| = |B|. But this does not imply that A = B. So |A| cannot just be A, taken in abstraction, unless that can identify distinct sets, ..or create new objects. | |
| From: William W. Tait (Frege versus Cantor and Dedekind [1996], V) | |
| A reaction: An elegant piece of argument, which shows rather crucial facts about abstraction. We are then obliged to ask how abstraction can create an object or a set, if the central activity of abstraction is just ignoring certain features. |