20 ideas
| 9766 | Study vagueness first by its logic, then by its truth-conditions, and then its metaphysics [Fine,K] |
| Full Idea: My investigation of vagueness began with the question 'What is the correct logic of vagueness?', which led to the further question 'What are the correct truth-conditions for a vague language?', which led to questions of meaning and existence. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: This is the most perfect embodiment of the strategy of analytical philosophy which I have ever read. It is the strategy invented by Frege in the 'Grundlagen'. Is this still the way to go, or has this pathway slowly sunk into the swamp? |
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
| From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 9775 | Excluded Middle, and classical logic, may fail for vague predicates [Fine,K] |
| Full Idea: Maybe classical logic fails for vagueness in Excluded Middle. If 'H bald ∨ ¬(H bald)' is true, then one disjunct is true. But if the second is true the first is false, and the sentence is either true or false, contrary to the borderline assumption. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 4) | |
| A reaction: Fine goes on to argue against the implication that we need a special logic for vague predicates. |
| 9771 | Logic holding between indefinite sentences is the core of all language [Fine,K] |
| Full Idea: If language is like a tree, then penumbral connection (logic holding among indefinite sentences) is the seed from which the tree grows, for it provides an initial repository of truths that are to be retained throughout all growth. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 2) | |
| A reaction: A nice incidental insight arising from his investigation of vagueness. People accept one another's reasons even when they are confused, or hopeless at expressing themselves. Nice. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
| From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
| A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 9768 | Vagueness is semantic, a deficiency of meaning [Fine,K] |
| Full Idea: I take vagueness to be a semantic feature, a deficiency of meaning. It is to be distinguished from generality, undecidability, and ambiguity. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: Sounds good. If we cut nature at the joints with our language, then nature is going to be too subtle and vast for our finite and gerrymandered language, and so it will break down in tricky situations. But maybe epistemology precedes semantics? |
| 9776 | A thing might be vaguely vague, giving us higher-order vagueness [Fine,K] |
| Full Idea: There is a possibility of 'higher-order vagueness'. The vague may be vague, or vaguely vague, and so on. If J has few hairs on his head than H, then he may be a borderline case of a borderline case. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 5) | |
| A reaction: Such slim grey areas can also be characterised as those where you think he is definitely bald, but I am not so sure. |
| 9767 | A vague sentence is only true for all ways of making it completely precise [Fine,K] |
| Full Idea: A vague sentence is (roughly stated) true if and only if it is true for all ways of making it completely precise (the 'super-truth theory'). | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], Intro) | |
| A reaction: Intuitively this sounds quite promising. Personally I think we should focus on the 'proposition' rather than the 'sentence' (where fifteen sentences might be needed before we can agree on the one proposition). |
| 9770 | Logical connectives cease to be truth-functional if vagueness is treated with three values [Fine,K] |
| Full Idea: With a three-value approach, if P is 'blob is pink' and R is 'blob is red', then P&P is indefinite, but P&R is false, and P∨P is indefinite, but P∨R is true. This means the connectives & and ∨ are not truth-functional. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 1) | |
| A reaction: The point is that there could then be no logic in any way classical for vague sentences and three truth values. A powerful point. |
| 9772 | Meaning is both actual (determining instances) and potential (possibility of greater precision) [Fine,K] |
| Full Idea: The meaning of an expression is the product of both its actual meaning (what helps determine its instances and counter-instances), and its potential meaning (the possibilities for making it more precise). | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 2) | |
| A reaction: A modal approach to meaning is gloriously original. Being quite a fan of real modalities (the possibilities latent in actuality), I find this intuitively appealing. |
| 9773 | With the super-truth approach, the classical connectives continue to work [Fine,K] |
| Full Idea: With the super-truth approach, if P is 'blob is pink' and R is 'blob is red', then P&R is false, and P∨R is true, since one of P and R is true and one is false in any complete and admissible specification. It encompasses all 'penumbral truths'. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 3) | |
| A reaction: [See Idea 9767 for the super-truth approach, and Idea 9770 for a contrasting view] The approach, which seems quite appealing, is that we will in no circumstances give up basic classical logic, but we will make maximum concessions to vagueness. |
| 9774 | Borderline cases must be under our control, as capable of greater precision [Fine,K] |
| Full Idea: Any borderline case must be under our control, in the sense that it can be settled by making the predicates more precise. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 3) | |
| A reaction: Sounds good. Consider an abstract concept like the equator. It is precise on a map of the world, but vague when you are in the middle of the tropics. But we can always form a committee to draw a (widish) line on the ground delineating it. |
| 9769 | Vagueness can be in predicates, names or quantifiers [Fine,K] |
| Full Idea: There are three possible sources of vagueness: the predicates, the names, and the quantifiers. | |
| From: Kit Fine (Vagueness, Truth and Logic [1975], 1) | |
| A reaction: Presumably a vagueness about the domain of discussion would be a vagueness in the quantifier. This is a helpful preliminary division, in the semantic approach to vagueness. |