21 ideas
| 18073 | Dummett says classical logic rests on meaning as truth, while intuitionist logic rests on assertability [Dummett, by Kitcher] |
| Full Idea: Dummett argues that classical logic depends on the choice of the concept of truth as central to the theory of meaning, while for the intuitionist the concept of assertability occupies this position. | |
| From: report of Michael Dummett (The philosophical basis of intuitionist logic [1973]) by Philip Kitcher - The Nature of Mathematical Knowledge 06.5 | |
| A reaction: Since I can assert any nonsense I choose, this presumably means 'warranted' assertability, which is tied to the concept of proof in mathematics. You can reason about falsehoods, or about uninterpreted variables. Can you 'assert' 'Fx'? |
| 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. |
| 9038 | We must distinguish what the speaker denotes by a name, from what the name denotes [Evans] |
| Full Idea: There are two related but distinguishable questions concerning proper names: what the speaker denotes (upon an occasion), and what the name denotes. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: I don't think any account of language makes sense without this sort of distinction, as in my favourite example: the password is 'swordfish'. So how does language gets its own meanings, independent of what speakers intend? |
| 5824 | How can an expression be a name, if names can change their denotation? [Evans] |
| Full Idea: We need an account of what makes an expression into a name for something that will allow names to change their denotations. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: Presumably an example would be 'The Prime Minister is in the building'. Evans proposes to discuss communication, rather than strict meanings and descriptions. |
| 9042 | A private intention won't give a name a denotation; the practice needs it to be made public [Evans] |
| Full Idea: Intentions alone don't bring it about that a name gets a denotation; without the intention being manifest there cannot be the common knowledge required for the practice. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: Well, I might have a private name for some hated colleague which I mutter to myself whenever I see her. The way names, and language generally, becomes ossified is by joining the great impersonal sea of the language. ..waves of bones, |
| 9041 | The Causal Theory of Names is wrong, since the name 'Madagascar' actually changed denotation [Evans] |
| Full Idea: Change of denotation is decisive against the Causal Theory of Names. Changes of denotation actually occur: a hearsay report misunderstood by Marco Polo transferred the name 'Madagascar' from a portion of the mainland to the African island. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: This doesn't sound decisive, as you could give an intermediate causal account of Marco Polo's mistake. I might take the famous name Winston, and baptise my son with it. And I might have done it because I thought Winston was a German dictator. |
| 19057 | Classical quantification is an infinite conjunction or disjunction - but you may not know all the instances [Dummett] |
| Full Idea: Classical quantification represents an infinite conjunction or disjunction, and the truth-value is determined by the infinite sum or product of the instances ....but this presupposes that all the instances already possess determinate truth-values. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.246) | |
| A reaction: In the case of the universal quantifier, Dummett is doing no more than citing the classic empiricism objection to induction - that you can't make the universal claim if you don't know all the instances. The claim is still meaningful, though. |
| 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. |
| 19055 | Stating a sentence's truth-conditions is just paraphrasing the sentence [Dummett] |
| Full Idea: An ability to state the condition for the truth of a sentence is, in effect, no more than an ability to express the content of the sentence in other words. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.224) | |
| A reaction: Alternatively, if you give something other than a paraphrase of the sentence as its meaning (such as a proof of its truth), then you seem to have departed from your target sentence. Can we reduce and eliminate our sentences in this way? |
| 19056 | If a sentence is effectively undecidable, we can never know its truth conditions [Dummett] |
| Full Idea: If a sentence is effectively undecidable, the condition which must obtain for it to be true is not one which we are capable of recognising whenever it obtains, or of getting ourselves in a position to do so. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.225) | |
| A reaction: The instances of 'undecidable' sentences are most clearly seen in mathematics, such as the Continuum Hypothesis or Goldbach's Conjecture, or anything involving vast infinite cardinals. But do you need precise truth-conditions for meaning? |
| 19054 | Meaning as use puts use beyond criticism, and needs a holistic view of language [Dummett] |
| Full Idea: If use constitutes meaning, it might seem that use is beyond criticism. ....But such an attitude can, ultimately, be supported onlly by the adoption of a holistic view of language. | |
| From: Michael Dummett (The philosophical basis of intuitionist logic [1973], p.218) | |
| A reaction: Dummett goes on to say that the rejection of the holistic view of mathematical meaning leads to his preference for intuitionistic logic. |
| 5825 | Speakers intend to refer to items that are the source of their information [Evans] |
| Full Idea: In general, a speaker intends to refer to the item that is the dominant source of his associated body of information. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: This sounds like a theory of reference which fully preserves the spirit of traditional empiricism. Speakers refer to ideas which connect to the source of their underlying impressions. |
| 5823 | The intended referent of a name needs to be the cause of the speaker's information about it [Evans] |
| Full Idea: A necessary (but not sufficient) condition for x's being the intended referent of S's use of a name is that x should be the source of the causal origin of the body of information that S has associated with the name. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: This is Evans's adaptation of Kripke's causal theory of names. This cries out for a counterexample. I say something about General Montgomery, having just listened to 'Monty's Double' give a talk, believing it was Montgomery? |
| 9039 | If descriptions are sufficient for reference, then I must accept a false reference if the descriptions fit [Evans] |
| Full Idea: The strong thesis (that descriptions are sufficient for reference) is outrageous. It would mean that if Mr X is wrongly introduced to me as Mr Y, then I truly say 'this is Mr Y' if X overwhelmingly satisfies descriptions of Y. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: [I omit some qualifying phrases] Evans says that probably no one ever held this view. It seems right. In the case of an electron it would seem that all the descriptions could be the same, except space-time location. Same electron as yesterday? |
| 9043 | We use expressions 'deferentially', to conform to the use of other people [Evans] |
| Full Idea: Sometimes we use expressions with the overriding intention to conform to the use made of them by some other person or persons. I shall say we use the expression 'deferentially'; examples might be 'viol' or 'minuet'. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §II) | |
| A reaction: I presume Evans wasn't very musical. This label sounds useful, if you wish to connect Grice's account of meaning with Putnam's externalist account of concepts, where deference to experts is crucial. Is all linguistic usage deferential? |
| 9040 | Charity should minimize inexplicable error, rather than maximising true beliefs [Evans] |
| Full Idea: I think the Principle of Charity (maximise true beliefs) is unacceptable. The acceptable principle enjoins minimizing the attribution of inexplicable error and cannot be operated without a theory of the causation of belief for the creatures investigated. | |
| From: Gareth Evans (The Causal Theory of Names [1973], §I) | |
| A reaction: The normal principle of charity certainly seems on shaky ground if you think you have encountered a fairly normal tribe, when they in fact are in possession of the weirdest belief system on the entire planet. |