18 ideas
| 21463 | Hamann, Herder and Jacobi were key opponents of the Enlightenment [Gardner] |
| Full Idea: Hamann, Herder and Jacobi are central figues in the reaction against Enlightenment. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10 'immediate') | |
| A reaction: From a British perspective I would see Hume as the leading such figure. Hamann emphasised the neglect of the role of language. Jacobi was a Christian. |
| 21459 | Kant halted rationalism, and forced empiricists to worry about foundations [Gardner] |
| Full Idea: Kant's Critique swiftly brought rationalism to a halt, and after Kant empiricism has displayed a nervousness regarding its foundations, and been forced to assume more sophisticated forms. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10 Intro) | |
| A reaction: See the ideas of Laurence Bonjour for a modern revival of rationalism. After Kant philosophers either went existential, or stared gloomily into the obscure depths. Formal logic was seen as a possible rope ladder down. |
| 21460 | Only Kant and Hegel have united nature, morals, politics, aesthetics and religion [Gardner] |
| Full Idea: Apart from Hegel, no later philosophical system equals in stature Kant's attempt to weld together the diverse fields of natural science, morality, politics, aesthetics and religion into a systematic overarching epistemological and metaphysical unity. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 10) | |
| A reaction: Earlier candidate are Plato and Aristotle. Earlier Enlightenment figures say little about morality or aesthetics. Hobbes ranges widely. Aquinas covered most things. |
| 21443 | Transcendental proofs derive necessities from possibilities (e.g. possibility of experiencing objects) [Gardner] |
| Full Idea: A transcendental proof converts a possibility into a necessity: by saying under what conditions experience of objects is possible, transcendental proofs show those conditions to be necessary for us to the extent that we have any experience of objects. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 02 'Transc') | |
| A reaction: They appear to be hypothetical necessities, rather than true metaphysical necessities. Gardner is discussing Kant, but seems to be generalising. Hypothetical necessities are easy: if it is flying, it is necessarily above the ground. |
| 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) |
| 21444 | Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner] |
| Full Idea: There is now 'pure' geometry, consisting of formal systems based on axioms for which truth is not claimed, and which are consequently not synthetic; and 'applied', a branch of physics, the truth of which is empirical, and therefore not a priori. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 03 'Maths') | |
| A reaction: His point is that there is no longer any room for a priori geometry. Might the same division be asserted of arithmetic, or analysis, or set theory? |
| 21453 | Leibnizian monads qualify as Kantian noumena [Gardner] |
| Full Idea: Leibnizian monads clearly satisfy Kant's definition of noumena. | |
| From: Sebastian Gardner (Kant and the Critique of Pure Reason [1999], 06 'Noumena') | |
| A reaction: This needs qualifying, because Leibniz clearly specifies the main attributes of monads, where Kant is adamant that we can saying virtually nothing about noumena. |
| 16463 | Adams says actual things have haecceities, but not things that only might exist [Adams,RM, by Stalnaker] |
| Full Idea: Adams favours haecceitism about actual things but no haecceities for things that might exist but don't. | |
| From: report of Robert Merrihew Adams (Actualism and Thisness [1981]) by Robert C. Stalnaker - Mere Possibilities 4.2 | |
| A reaction: This contrasts with Plantinga, who proposes necessary essences for everything, even for what might exist. Plantinga sounds crazy to me, Adams merely interesting but not too plausible. |