16 ideas
| 17963 | The facts of geometry, arithmetic or statics order themselves into theories [Hilbert] |
| Full Idea: The facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, electrodynamics into a theory of statics, electrodynamics, or the facts of the physics of gases into a theory of gases. | |
| From: David Hilbert (Axiomatic Thought [1918], [03]) | |
| A reaction: This is the confident (I would say 'essentialist') view of axioms, which received a bit of a setback with Gödel's Theorems. I certainly agree that the world proposes an order to us - we don't just randomly invent one that suits us. |
| 17966 | Axioms must reveal their dependence (or not), and must be consistent [Hilbert] |
| Full Idea: If a theory is to serve its purpose of orienting and ordering, it must first give us an overview of the independence and dependence of its propositions, and second give a guarantee of the consistency of all of the propositions. | |
| From: David Hilbert (Axiomatic Thought [1918], [09]) | |
| A reaction: Gödel's Second theorem showed that the theory can never prove its own consistency, which made the second Hilbert requirement more difficult. It is generally assumed that each of the axioms must be independent of the others. |
| 17967 | To decide some questions, we must study the essence of mathematical proof itself [Hilbert] |
| Full Idea: It is necessary to study the essence of mathematical proof itself if one wishes to answer such questions as the one about decidability in a finite number of operations. | |
| From: David Hilbert (Axiomatic Thought [1918], [53]) |
| 17965 | The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert] |
| Full Idea: The linearity of the equation of the plane and of the orthogonal transformation of point-coordinates is completely adequate to produce the whole broad science of spatial Euclidean geometry purely by means of analysis. | |
| From: David Hilbert (Axiomatic Thought [1918], [05]) | |
| A reaction: This remark comes from the man who succeeded in producing modern axioms for geometry (in 1897), so he knows what he is talking about. We should not be wholly pessimistic about Hilbert's ambitious projects. He had to dig deeper than this idea... |
| 17964 | Number theory just needs calculation laws and rules for integers [Hilbert] |
| Full Idea: The laws of calculation and the rules of integers suffice for the construction of number theory. | |
| From: David Hilbert (Axiomatic Thought [1918], [05]) | |
| A reaction: This is the confident Hilbert view that the whole system can be fully spelled out. Gödel made this optimism more difficult. |
| 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. |
| 7439 | The qualities involved in sensations are entirely intentional [Anscombe, by Armstrong] |
| Full Idea: Anscombe argued that the qualities involved in sensations are one and all intentional only (and I think this holds even for the bodily sensations). | |
| From: report of G.E.M. Anscombe (The Intentionality of Sensation [1965]) by David M. Armstrong - Pref to new 'Materialist Theory' p.xxii | |
| A reaction: Compare Harry Gildersleve's exactly opposite proposal in Idea 7272. I think I am coming round to the Anscombe view, which builds the more mysterious up from the less mysterious. Gildersleve must explain how atomic qualia arise. |
| 17968 | By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge [Hilbert] |
| Full Idea: By pushing ahead to ever deeper layers of axioms ...we also win ever-deeper insights into the essence of scientific thought itself, and become ever more conscious of the unity of our knowledge. | |
| From: David Hilbert (Axiomatic Thought [1918], [56]) | |
| A reaction: This is the less fashionable idea that scientific essentialism can also be applicable in the mathematic sciences, centring on the project of axiomatisation for logic, arithmetic, sets etc. |