15 ideas
| 8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
| Full Idea: Maddy dispenses with pure sets, by sketching a strong set theory in which everything is either a physical object or a set of sets of ...physical objects. Eventually a physiological story of perception will extend to sets of physical objects. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: This doesn't seem to find many supporters, but if we accept the perception of resemblances as innate (as in Hume and Quine), it is isn't adding much to see that we intrinsically see things in groups. |
| 10718 | A natural number is a property of sets [Maddy, by Oliver] |
| Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990], 3 §2) by Alex Oliver - The Metaphysics of Properties | |
| A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things. |
| 8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
| Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
| A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things. |
| 17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
| Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience. | |
| From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5 | |
| A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory. |
| 12184 | Logical necessity overrules all other necessities [McFetridge] |
| Full Idea: If it is logically necessary that if p then q, then there is no other sense of 'necessary' in which it is not necessary that if p then q. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
| A reaction: The thesis which McFetridge proposes to defend. The obvious rival would be metaphysical necessity, and the rival claim would presumably be that things are only logically necessary if that is entailed by a metaphysical necessity. Metaphysics drives logic. |
| 15083 | The fundamental case of logical necessity is the valid conclusion of an inference [McFetridge, by Hale] |
| Full Idea: McFetridge's conception of logical necessity is one which sees the concept as receiving its fundamental exemplification in the connection between the premiss and conclusion of a deductively valid inference. | |
| From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
| A reaction: This would mean that p could be logically necessary but false (if it was a valid argument from false premisses). What if it was a valid inference in a dodgy logical system (including 'tonk', for example)? |
| 15084 | In the McFetridge view, logical necessity means a consequent must be true if the antecedent is [McFetridge, by Hale] |
| Full Idea: McFetridge's view proves that if the conditional corresponding to a valid inference is logically necessary, then there is no sense in which it is possible that its antecedent be true but its consequent false. ..This result generalises to any statement. | |
| From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
| A reaction: I am becoming puzzled by Hale's assertion that logical necessity is 'absolute', while resting his case on a conditional. Are we interested in the necessity of the inference, or the necessity of the consequent? |
| 12180 | Logical necessity requires that a valid argument be necessary [McFetridge] |
| Full Idea: There will be a legitimate notion of 'logical' necessity only if there is a notion of necessity which attaches to the claim, concerning a deductively valid argument, that if the premisses are true then so is the conclusion. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
| A reaction: He quotes Aristotle's Idea 11148 in support. Is this resting a stronger idea on a weaker one? Or is it the wrong way round? We endorse validity because we see the necessity; we don't endorse necessity because we see 'validity'. |
| 12181 | Traditionally, logical necessity is the strongest, and entails any other necessities [McFetridge] |
| Full Idea: The traditional crucial assumption is that logical necessity is the strongest notion of necessity. If it is logically necessary that p, then it is necessary that p in any other use of the notion of necessity there may be (physically, practically etc.). | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
| A reaction: Sounds right. We might say it is physically necessary simply because it is logically necessary, and even that it is metaphysically necessary because it is logically necessary (required by logic). Logical possibility is hence the weakest kind? |
| 12183 | It is only logical necessity if there is absolutely no sense in which it could be false [McFetridge] |
| Full Idea: Is there any sense in which, despite an ascription of necessity to p, it is held that not-p is possible? If there is, then the original claim then it was necessary is not a claim of 'logical' necessity (which is the strongest necessity). | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
| A reaction: See Idea 12181, which leads up to this proposed "test" for logical necessity. McFetridge has already put epistemic ('for all I know') possibility to one side. □p→¬◊¬p is the standard reading of necessity. His word 'sense' bears the burden. |
| 12192 | The mark of logical necessity is deduction from any suppositions whatever [McFetridge] |
| Full Idea: The manifestation of the belief that a mode of inference is logically necessarily truth-preserving is the preparedness to employ that mode of inference in reasoning from any set of suppositions whatsoever. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §4) | |
| A reaction: He rests this on the idea of 'cotenability' of the two sides of a counterfactual (in Mill, Goodman and Lewis). There seems, at first blush, to be a problem of the relevance of the presuppositions. |
| 12182 | We assert epistemic possibility without commitment to logical possibility [McFetridge] |
| Full Idea: Time- and person-relative epistemic possibility can be asserted even when logical possibility cannot, such as undecided mathematical propositions. 'It may be that p' just comes to 'For all I know, not-p'. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
| A reaction: If it is possible 'for all I know', then it could be actual for all I know, and if we accept that it might be actual, we could hardly deny that it is logically possible. Logical and epistemic possibilities of mathematical p stand or fall together. |
| 12187 | Objectual modal realists believe in possible worlds; non-objectual ones rest it on the actual world [McFetridge] |
| Full Idea: The 'objectual modal realist' holds that what makes modal beliefs true are certain modal objects, typically 'possible worlds'. ..The 'non-objectual modal realist' says modal judgements are made true by how things stand with respect to this world. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
| A reaction: I am an enthusiastic 'non-objectual modal realist'. I accept the argument that real possible worlds have no relevance to the actual world, and explain nothing (see Jubien). The possibilities reside in the 'powers' of this world. See Molnar on powers. |
| 12186 | Modal realists hold that necessities and possibilities are part of the totality of facts [McFetridge] |
| Full Idea: The 'modal realist' holds that part of the totality of what is the case, the totality of facts, are such things as that certain events could have happened, certain propositions are necessarily true, if this happened then that would have been the case. | |
| From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
| A reaction: I am an enthusiastic modal realist. If the aim of philosophy is 'to understand' (and I take that to be the master idea of the subject) then no understanding is possible which excludes the possibilities and necessities in things. |
| 22591 | We know perfection when we see what is imperfect [Murdoch] |
| Full Idea: We know of perfection as we look upon what is imperfect. | |
| From: Iris Murdoch (Metaphysics as a Guide to Morals [1992], 13) | |
| A reaction: This is in the context of a discussion of the ontological argument for God's existence, but I seize on it as a nice expression of the idealisation capacity of our minds. The alternative is that perfection is innate idea, since we aren't seeing it. |