16 ideas
13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
14193 | 'Substance theorists' take modal properties as primitive, without structure, just falling under a sortal [Paul,LA] |
Full Idea: Some deep essentialists resist the need to explain the structure under de re modal properties, taking them as primitive. One version (which we can call 'substance theory') takes them to fall under a sortal concept, with no further explanation. | |
From: L.A. Paul (In Defense of Essentialism [2006], §1) | |
A reaction: A very helpful identification of what Wiggins stands for, and why I disagree with him. The whole point of essences is to provide a notion that fits in with sciences, which means they must have an explanatory role, which needs structures. |
14195 | If an object's sort determines its properties, we need to ask what determines its sort [Paul,LA] |
Full Idea: If the substance essentialist holds that the sort an object belongs to determines its de re modal properties (rather than the other way round), then he needs to give an (ontological, not conceptual) explanation of what determines an object's sort. | |
From: L.A. Paul (In Defense of Essentialism [2006], §1) | |
A reaction: See Idea 14193 for 'substance essentialism'. I find it quite incredible that anyone could think that a thing's sort could determine its properties, rather than the other way round. Even if sortals are conventional, they are not arbitrary. |
14196 | Substance essentialism says an object is multiple, as falling under various different sortals [Paul,LA] |
Full Idea: The explanation of material constitution given by substance essentialism is that there are multiple objects. A person is essentially human-shaped (falling under the human sort), while their hunk of tissue is accidentally human-shaped (as tissue). | |
From: L.A. Paul (In Defense of Essentialism [2006], §1) | |
A reaction: At this point sortal essentialism begins to look crazy. Persons are dubious examples (with sneaky dualism involved). A bronze statue is essentially harder to dent than a clay one, because of its bronze. If you remake it of clay, it isn't the same statue. |
14198 | Absolutely unrestricted qualitative composition would allow things with incompatible properties [Paul,LA] |
Full Idea: Absolutely unrestricted qualitative composition would imply that objects with incompatible properties and objects such as winged pigs or golden mountains were actual. | |
From: L.A. Paul (In Defense of Essentialism [2006], §5) | |
A reaction: Note that this is 'qualitative' composition, and not composition of parts. The objection seems to rule out unrestricted qualitative composition, since you could hardly combine squareness with roundness. |
14190 | Deep essentialist objects have intrinsic properties that fix their nature; the shallow version makes it contextual [Paul,LA] |
Full Idea: Essentialism says that objects have their properties essentially. 'Deep' essentialists take the (nontrivial) essential properties of an object to determine its nature. 'Shallow' essentialists substitute context-dependent truths for the independent ones. | |
From: L.A. Paul (In Defense of Essentialism [2006], Intro) | |
A reaction: If the deep essence determines a things nature, we should not need to say 'nontrivial'. This is my bete noire, the confusion of essential properties with necessary ones, where necessary properties (or predicates, at least) can indeed be trivial. |
14191 | Deep essentialists say essences constrain how things could change; modal profiles fix natures [Paul,LA] |
Full Idea: The deep essentialist holds that most objects have essential properties such that there are many ways they could not be, or many changes through which they could not persist. Objects' modal profiles characterize their natures. | |
From: L.A. Paul (In Defense of Essentialism [2006], Intro) | |
A reaction: This is the view I like, especially the last bit. If your modal profile doesn't determine your nature, then what does? Think of how you sum up a person at a funeral. Your modal profile is determined by dispositions and powers. |
14192 | Essentialism must deal with charges of arbitrariness, and failure to reduce de re modality [Paul,LA] |
Full Idea: Two objections to deep essentialism are that it falters when faced with a skeptical objection concerning arbitrariness, and the need for a reductive account of de re modality. | |
From: L.A. Paul (In Defense of Essentialism [2006], Intro) | |
A reaction: An immediate response to the second objection might be to say that modal facts about things are not reducible. The charge of arbitrariness (i.e. total arbitrariness, not just a bit of uncertainty) is the main thing a theory of essences must deal with. |
14197 | An object's modal properties don't determine its possibilities [Paul,LA] |
Full Idea: I reject the view that an object's de re modal properties determine its relations to possibilia. | |
From: L.A. Paul (In Defense of Essentialism [2006], §3) | |
A reaction: You'll have to read Paul to see why, but I flat disagree with her on this. The whole point of accepting such properties is to determine the modal profile of the thing, and hence see how it can fit into and behave in the world. |
14189 | 'Modal realists' believe in many concrete worlds, 'actualists' in just this world, 'ersatzists' in abstract other worlds [Paul,LA] |
Full Idea: A 'modal realist' believes that there are many concrete worlds, while the 'actualist' believes in only one concrete world, the actual world. The 'ersatzist' is an actualist who takes nonactual possible worlds and their contents to be abstracta. | |
From: L.A. Paul (In Defense of Essentialism [2006], Intro) | |
A reaction: My view is something like that modal realism is wrong, and actualism is right, and possible worlds (if they really are that useful) are convenient abstract fictions, constructed (if we have any sense) out of the real possibilities in the actual world. |
12900 | How could 'S knows he has hands' not have a fixed content? [Bach] |
Full Idea: How can it be that a sentence like 'George knows that he has hands', even with time and references fixed, does not have a fixed propositional content? | |
From: Kent Bach (The Emperor's New 'Knows' [2005], I) | |
A reaction: The appeal is to G.E. Moore's common sense view of immediate knowledge (Idea 6349). The reply is simply that the word 'knows' shifts its meaning, having high standards in sceptical philosophy classes, and low standards on the street. |
12901 | If contextualism is right, knowledge sentences are baffling out of their context [Bach] |
Full Idea: Contextualism seems to predict that if you encounter a knowledge attribution out of context you won't be in a position to grasp which proposition the sentence expresses. | |
From: Kent Bach (The Emperor's New 'Knows' [2005], I) | |
A reaction: It is only the word 'knows' which is at issue in the sentence. If someone is said to 'know' about the world of the fairies, we might well be puzzled as to what proposition was being expressed. Is the word 'flat' baffling out of context? |
12902 | Sceptics aren't changing the meaning of 'know', but claiming knowing is tougher than we think [Bach] |
Full Idea: When a sceptic brings up far-fetched possibilities and argues that we can't rule them out, he is not raising the standard for the word 'know'. He is showing it is tougher than we realise for a belief to qualify as normal knowledge at all. | |
From: Kent Bach (The Emperor's New 'Knows' [2005], III) | |
A reaction: [Bach cites Richard Feldman for this idea] I think that what happens in the contextual account is that 'true', 'belief' and 'know' retain their standard meaning, and it is 'justified' which shifts. 'I am fully justified' can have VERY different meanings! |