20 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. |
10017 | Truth in a model is more tractable than the general notion of truth [Hodes] |
Full Idea: Truth in a model is interesting because it provides a transparent and mathematically tractable model - in the 'ordinary' rather than formal sense of the term 'model' - of the less tractable notion of truth. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
A reaction: This is an important warning to those who wish to build their entire account of truth on Tarski's rigorously formal account of the term. Personally I think we should start by deciding whether 'true' can refer to the mental state of a dog. I say it can. |
10018 | Truth is quite different in interpreted set theory and in the skeleton of its language [Hodes] |
Full Idea: There is an enormous difference between the truth of sentences in the interpreted language of set theory and truth in some model for the disinterpreted skeleton of that language. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.132) | |
A reaction: This is a warning to me, because I thought truth and semantics only entered theories at the stage of 'interpretation'. I must go back and get the hang of 'skeletal' truth, which sounds rather charming. [He refers to set theory, not to logic.] |
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! |
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? |
10015 | Higher-order logic may be unintelligible, but it isn't set theory [Hodes] |
Full Idea: Brand higher-order logic as unintelligible if you will, but don't conflate it with set theory. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
A reaction: [he gives Boolos 1975 as a further reference] This is simply a corrective, because the conflation of second-order logic with set theory is an idea floating around in the literature. |
10011 | Identity is a level one relation with a second-order definition [Hodes] |
Full Idea: Identity should he considered a logical notion only because it is the tip of a second-order iceberg - a level 1 relation with a pure second-order definition. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) |
10016 | When an 'interpretation' creates a model based on truth, this doesn't include Fregean 'sense' [Hodes] |
Full Idea: A model is created when a language is 'interpreted', by assigning non-logical terms to objects in a set, according to a 'true-in' relation, but we must bear in mind that this 'interpretation' does not associate anything like Fregean senses with terms. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.131) | |
A reaction: This seems like a key point (also made by Hofweber) that formal accounts of numbers, as required by logic, will not give an adequate account of the semantics of number-terms in natural languages. |
10027 | Mathematics is higher-order modal logic [Hodes] |
Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic? |
10026 | Arithmetic must allow for the possibility of only a finite total of objects [Hodes] |
Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148) | |
A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system. |
10021 | It is claimed that numbers are objects which essentially represent cardinality quantifiers [Hodes] |
Full Idea: The mathematical object-theorist says a number is an object that represents a cardinality quantifier, with the representation relation as the entire essence of the nature of such objects as cardinal numbers like 4. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984]) | |
A reaction: [compressed] This a classic case of a theory beginning to look dubious once you spell it our precisely. The obvious thought is to make do with the numerical quantifiers, and dispense with the objects. Do other quantifiers need objects to support them? |
10022 | Numerical terms can't really stand for quantifiers, because that would make them first-level [Hodes] |
Full Idea: The dogmatic Frege is more right than wrong in denying that numerical terms can stand for numerical quantifiers, for there cannot be a language in which object-quantifiers and objects are simultaneously viewed as level zero. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.142) | |
A reaction: Subtle. We see why Frege goes on to say that numbers are level zero (i.e. they are objects). We are free, it seems, to rewrite sentences containing number terms to suit whatever logical form appeals. Numbers are just quantifiers? |
10023 | Talk of mirror images is 'encoded fictions' about real facts [Hodes] |
Full Idea: Talk about mirror images is a sort of fictional discourse. Statements 'about' such fictions are not made true or false by our whims; rather they 'encode' facts about the things reflected in mirrors. | |
From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.146) | |
A reaction: Hodes's proposal for how we should view abstract objects (c.f. Frege and Dummett on 'the equator'). The facts involved are concrete, but Hodes is offering 'encoding fictionalism' as a linguistic account of such abstractions. He applies it to numbers. |
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. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |