17 ideas
| 10825 | The notion of truth is to help us make use of the utterances of others [Field,H] |
| Full Idea: I suspect that the original purpose of the notion of truth was to aid us in utilizing the utterances of others in drawing conclusions about the world,...so we must attend to its social role, and that being in a position to assert something is what counts. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], §5) | |
| A reaction: [Last bit compressed] This sounds excellent. Deflationary and redundancy views are based on a highly individualistic view of utterances and truth, but we need to be much more contextual and pragmatic if we are to get the right story. |
| 10820 | In the early 1930s many philosophers thought truth was not scientific [Field,H] |
| Full Idea: In the early 1930s many philosophers believed that the notion of truth could not be incorporated into a scientific conception of the world. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], §3) | |
| A reaction: This leads on to an account of why Tarski's formal version was so important, and Field emphasises Tarski's physicalist metaphysic. |
| 13499 | Tarski reduced truth to reference or denotation [Field,H, by Hart,WD] |
| Full Idea: Tarski can be viewed as having reduced truth to reference or denotation. | |
| From: report of Hartry Field (Tarski's Theory of Truth [1972]) by William D. Hart - The Evolution of Logic 4 |
| 10818 | Tarski really explained truth in terms of denoting, predicating and satisfied functions [Field,H] |
| Full Idea: A proper account of Tarski's truth definition explains truth in terms of three other semantic notions: what it is for a name to denote something, and for a predicate to apply to something, and for a function symbol to be fulfilled by a pair of things. | |
| From: Hartry Field (Tarski's Theory of Truth [1972]) | |
| A reaction: This is Field's 'T1' version, which is meant to spell out what was really going on in Tarski's account. |
| 10817 | Tarski just reduced truth to some other undefined semantic notions [Field,H] |
| Full Idea: It is normally claimed that Tarski defined truth using no undefined semantic terms, but I argue that he reduced the notion of truth to certain other semantic notions, but did not in any way explicate these other notions. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], §0) |
| 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. |
| 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. |
| 10819 | Tarski gives us the account of truth needed to build a group of true sentences in a model [Field,H] |
| Full Idea: Model theory must choose the denotations of the primitives so that all of a group of sentences come out true, so we need a theory of how the truth value of a sentence depends on the denotation of its primitive nonlogical parts, which Tarski gives us. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], §1) |
| 10827 | Model theory is unusual in restricting the range of the quantifiers [Field,H] |
| Full Idea: In model theory we are interested in allowing a slightly unusual semantics for quantifiers: we are willing to allow that the quantifier not range over everything. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], n 5) |
| 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) |
| 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. |
| 18431 | Internal relations combine some tropes into a nucleus, which bears the non-essential tropes [Simons, by Edwards] |
| Full Idea: Simons's 'nuclear' option blends features of the substratum and bundle theories. First we have tropes collected by virtue of their internal relations, forming the essential kernel or nucleus. This nucleus then bears the non-essential tropes. | |
| From: report of Peter Simons (Particulars in Particular Clothing [1994], p.567) by Douglas Edwards - Properties 3.5 | |
| A reaction: [compression of Edwards's summary] This strikes me as being a remarkably good theory. I am not sure of the ontological status of properties, such that they can (unaided) combine to make part of an object. What binds the non-essentials? |
| 10826 | 'Valence' and 'gene' had to be reduced to show their compatibility with physicalism [Field,H] |
| Full Idea: 'Valence' and 'gene' were perfectly clear long before anyone succeeded in reducing them, but it was their reducibility and not their clarity before reduction that showed them to be compatible with physicalism. | |
| From: Hartry Field (Tarski's Theory of Truth [1972], §5) |
| 7615 | Field says reference is a causal physical relation between mental states and objects [Field,H, by Putnam] |
| Full Idea: In Field's view reference is a 'physicalistic relation', i.e. a complex causal relation between words or mental representations and objects or sets of objects; it is up to physical science to discover what that physicalistic relation is. | |
| From: report of Hartry Field (Tarski's Theory of Truth [1972]) by Hilary Putnam - Reason, Truth and History Ch.2 | |
| A reaction: I wouldn't hold your breath while the scientists do their job. If physicalism is right then Field is right, but physics seems no more appropriate for giving a theory of reference than it does for giving a theory of music. |