22 ideas
| 16567 | Scientists know everything about nothing, philosophers nothing about everything [Sagan,D] |
| Full Idea: The scientist learns more and more about less and less, until she knows everything about nothing, whereas a philosopher learns less and less about more and more until he knows nothing about everything. | |
| From: Dorion Sagan (Cosmic Apprentice [2013]) | |
| A reaction: [Came via Twitter] Not sure if this is true, but it is too nice to miss. |
| 13030 | Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen] |
| Full Idea: Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13032 | Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen] |
| Full Idea: Axiom of Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z). Any pair of entities must form a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) | |
| A reaction: Repeated applications of this can build the hierarchy of sets. |
| 13033 | Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A) [Kunen] |
| Full Idea: Axiom of Union: ∀F ∃A ∀Y ∀x (x ∈ Y ∧ Y ∈ F → x ∈ A). That is, the union of a set (all the members of the members of the set) must also be a set. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13037 | Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x) [Kunen] |
| Full Idea: Axiom of Infinity: ∃x (0 ∈ x ∧ ∀y ∈ x (S(y) ∈ x). That is, there is a set which contains zero and all of its successors, hence all the natural numbers. The principal of induction rests on this axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §1.7) |
| 13038 | Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen] |
| Full Idea: Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}. | |
| From: Kenneth Kunen (Set Theory [1980], §1.10) |
| 13034 | Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen] |
| Full Idea: Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13039 | Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen] |
| Full Idea: Axiom of Foundation: ∀x (∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))). Aka the 'Axiom of Regularity'. Combined with Choice, it means there are no downward infinite chains. | |
| From: Kenneth Kunen (Set Theory [1980], §3.4) |
| 13036 | Choice: ∀A ∃R (R well-orders A) [Kunen] |
| Full Idea: Axiom of Choice: ∀A ∃R (R well-orders A). That is, for every set, there must exist another set which imposes a well-ordering on it. There are many equivalent versions. It is not needed in elementary parts of set theory. | |
| From: Kenneth Kunen (Set Theory [1980], §1.6) |
| 13029 | Set Existence: ∃x (x = x) [Kunen] |
| Full Idea: Axiom of Set Existence: ∃x (x = x). This says our universe is non-void. Under most developments of formal logic, this is derivable from the logical axioms and thus redundant, but we do so for emphasis. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) |
| 13031 | Comprehension: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ) [Kunen] |
| Full Idea: Comprehension Scheme: for each formula φ without y free, the universal closure of this is an axiom: ∃y ∀x (x ∈ y ↔ x ∈ z ∧ φ). That is, there must be a set y if it can be defined by the formula φ. | |
| From: Kenneth Kunen (Set Theory [1980], §1.5) | |
| A reaction: Unrestricted comprehension leads to Russell's paradox, so restricting it in some way (e.g. by the Axiom of Specification) is essential. |
| 13040 | Constructibility: V = L (all sets are constructible) [Kunen] |
| Full Idea: Axiom of Constructability: this is the statement V = L (i.e. ∀x ∃α(x ∈ L(α)). That is, the universe of well-founded von Neumann sets is the same as the universe of sets which are actually constructible. A possible axiom. | |
| From: Kenneth Kunen (Set Theory [1980], §6.3) |
| 3523 | Shadows are supervenient on their objects, but not reducible [Maslin] |
| Full Idea: Shadows are distinct from the physical objects casting the shadows and irreducible to them; any attempt at reduction would be incoherent, as it would entail identifying a shadow with the object of which it is a shadow. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 6.3) | |
| A reaction: Another failure to find a decent analogy for what is claimed in property dualism. A 'shadow' is a reification of the abstract concept of an absence of light. Objects lose their shadows at dusk, but the object itself doesn't change. |
| 3517 | 'Ontology' means 'study of things which exist' [Maslin] |
| Full Idea: The word 'ontology' is derived from the Greek word 'ontia', which means 'things which exist'. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 1.1) |
| 3538 | Analogy to other minds is uncheckable, over-confident and chauvinistic [Maslin] |
| Full Idea: The argument from analogy makes it impossible to check my inductive inferences because of the privacy of other minds; it also seems irresponsible to generalise from a single case; and it seems like a case of human chauvinism. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 8.2) | |
| A reaction: Privacy of other minds need not imply scepticism about them. I'm a believer, so I have no trouble checking my theories. Solipsists can't 'check' anything. It isn't 'irresponsible' to generalise from one case if that is all you have. |
| 3540 | If we are brains then we never meet each other [Maslin] |
| Full Idea: If I am my brain this leads to the odd result that you have never met me because you have never seen my brain. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 10.7) | |
| A reaction: 'Star Trek' is full of aliens who appear beautiful, and turn out to be ugly grey lumps. 'I am my face' would be just as odd, particularly if I were in a coma, or dead. |
| 3518 | I'm not the final authority on my understanding of maths [Maslin] |
| Full Idea: I may be the final authority on whether my shoe pinches, but I am manifestly not the final authority on whether I understand some mathematical theorem. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 1.7) | |
| A reaction: However, it doesn't follow that his teachers are the final authority either, because he may get correct answers by an algorithm, and bluff his way when demonstrating his understanding. Who knows whether anyone really understands anything? |
| 3530 | Denial of purely mental causation will lead to epiphenomenalism [Maslin] |
| Full Idea: If mental events are causally efficacious only by virtue of their physical features and not their mental ones, …then anomalous monism leads straight to ephiphenomenalism. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 7.6) | |
| A reaction: As epiphenomenalism strikes me as being incoherent (see Idea 7379), what this amounts to is that either mental effects are causally efficacious, or they are not worth mentioning. I take them to be causally efficacious because they are brain events. |
| 3520 | Token-identity removes the explanatory role of the physical [Maslin] |
| Full Idea: In token-identity mental and physical features seem as unrelated as colour and shape, which is very weak physicalism because it does not allow physical states an explanatory role in accounting for mental states. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 3.8.6) | |
| A reaction: Colour and shape are not totally unrelated, as they can both be totally explained by a full knowledge of the physical substance involved. ...But maybe if we fully understood Spinoza's single substance...? See Idea 4834. |
| 3528 | Causality may require that a law is being followed [Maslin] |
| Full Idea: The principle of nomological causality says that if two events are intrinsically causally related, there must be a strict physical law under which they can be subsumed. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 7.5) |
| 3525 | Strict laws make causation logically necessary [Maslin] |
| Full Idea: 'Deductive-nomological' explanation consists of two premises - a strict law with no exceptions and supporting deterministic counterfactuals, and a statement of an event which falls under the law - which together logically require the effect. | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 7.4) |
| 3527 | Strict laws allow no exceptions and are part of a closed system [Maslin] |
| Full Idea: 'Strict' laws of nature contain no ceteris paribus clauses ('all things being equal'), and are part of a closed system (so that whatever affects the system must be included within the system). | |
| From: Keith T. Maslin (Introduction to the Philosophy of Mind [2001], 7.5) |