34 ideas
| 12442 | 'Mickey Mouse is a fictional mouse' is true without a truthmaker [Azzouni] |
| Full Idea: 'Mickey Mouse is a fictional mouse' can be taken as true without have any truthmaker. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: There might be an equivocation over 'true' here. 'What, really really true that he IS a fictional mouse?' |
| 12439 | Truth is dispensable, by replacing truth claims with the sentence itself [Azzouni] |
| Full Idea: No truth predicate is ever indispensable, because Tarski biconditionals, the equivalences between sentences and explicit truth ascriptions to those sentences, allow us to replace explicit truth ascriptions with the sentences themselves. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.1) | |
| A reaction: Holding a sentence to be true isn't the same as saying that it is true, and it isn't the same as saying the sentence, because one might say it in an ironic tone of voice. |
| 12437 | Truth lets us assent to sentences we can't explicitly exhibit [Azzouni] |
| Full Idea: My take on truth is a fairly deflationary one: The role of the truth predicate is to enable us to assent to sentences we can't explicitly exhibit. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Intro) | |
| A reaction: Clearly this is a role for truth, as in 'I forget what he said, but I know it was true', but it isn't remotely what most people understand by true. We use 'true' about totally explicit sentences all the time. |
| 12452 | Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer] |
| Full Idea: Not to the mathematician, but to the psychologist, belongs the task of explaining why ...we are averse to so-called contradictory systems in which the negative as well as the positive of certain propositions are valid. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.79) | |
| A reaction: Was the turning point of Graham Priest's life the day he read this sentence? I don't agree. I take the principle of non-contradiction to be a highly generalised observation of how the world works (and Russell agrees with me). |
| 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) |
| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |
| Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth. | |
| From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2 | |
| A reaction: [Brouwer 1952:510-11] |
| 12446 | Names function the same way, even if there is no object [Azzouni] |
| Full Idea: Names function the same way (semantically and grammatically) regardless of whether or not there's an object that they refer to. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3 n55) | |
| A reaction: I take this to be a fairly clear rebuttal of the 'Fido'-Fido view of names (that the meaning of the name IS the dog), which never seems to quite go away. A name is a peg on which description may be hung, seems a good slogan to me. |
| 18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
| Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking! |
| 18247 | Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro] |
| Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities. | |
| From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6 | |
| A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer. |
| 12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
| Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77) | |
| A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'. |
| 18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
| Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses. |
| 12454 | Intuitionists only accept denumerable sets [Brouwer] |
| Full Idea: The intuitionist recognises only the existence of denumerable sets. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days. |
| 12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
| Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals. |
| 8728 | Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer] |
| Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths. | |
| From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2 | |
| A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time! |
| 12447 | That all existents have causal powers is unknowable; the claim is simply an epistemic one [Azzouni] |
| Full Idea: If the argument isn't that, metaphysically speaking, anything that exists must have causal powers - how on earth would we show that? - rather, the claim is an epistemic one. Any thing we're in a position to know about we must causally interact with. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.4) | |
| A reaction: A very good point. I am attracted to causal power as a criterion for existence, but Azzouni's distinction is vital. Maybe there is just no point in even talking about things which exist but have no causal powers. |
| 12445 | If fictional objects really don't exist, then they aren't abstract objects [Azzouni] |
| Full Idea: It's robustly part of common sense that fictional objects don't exist in any sense at all, and this means they aren't abstracta either. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: Nice. It is so easy to have some philosopher dilute and equivocate over the word 'object' until you find yourself committed to all sorts of daft things as somehow having objectual existence. We can discuss things which don't exist in any way at all. |
| 12449 | Modern metaphysics often derives ontology from the logical forms of sentences [Azzouni] |
| Full Idea: It is widespread in contemporary metaphysics to extract commitments to various types of object on the basis of the logical form of certain sentences. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.7) | |
| A reaction: I'm with Azzouni in thinking that this procedure is a very bad idea. I'm increasingly inclined towards the wild view that people are only ontologically committed to things if they explicitly say that they are so committed. |
| 12440 | If objectual quantifiers ontologically commit, so does the metalanguage for its semantics [Azzouni] |
| Full Idea: The argument that objectual quantifiers are ontologically committing has the crucial and unnoticed presupposition that the language in which the semantics for the objectual quantifiers is couched (the 'metalanguage') also has quantifiers with commitment. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: That is, presumably we find ourselves ontologically committed to the existence of quantifiers, and are also looking at an infinite regress. See Idea 12439. |
| 12438 | In the vernacular there is no unequivocal ontological commitment [Azzouni] |
| Full Idea: There are no linguistic devices, no idioms (not 'there is', not 'exists') that unequivocally indicate ontological commitment in the vernacular. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Intro) | |
| A reaction: This seems right, since people talk in such ways about soap opera, while understanding the ontological situation perfectly well. Presumably Quine is seeking higher standards than the vernacular, if we are doing science. |
| 12441 | We only get ontology from semantics if we have already smuggled it in [Azzouni] |
| Full Idea: A slogan: One can't read ontological commitments from semantic conditions unless one has already smuggled into those semantic conditions the ontology one would like to read off. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.3) | |
| A reaction: The arguments supporting this are subtle, but it's good enough for me, as I never thought anyone was ontologically committed just because they used the vagueries of language to try to say what's going on around here. |
| 18465 | An 'equivalence' relation is one which is reflexive, symmetric and transitive [Kunen] |
| Full Idea: R is an equivalence relation on A iff R is reflexive, symmetric and transitive on A. | |
| From: Kenneth Kunen (The Foundations of Mathematics (2nd ed) [2012], I.7.1) |
| 12448 | Things that don't exist don't have any properties [Azzouni] |
| Full Idea: Things that don't exist don't have any properties. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.4) | |
| A reaction: Sounds reasonable! I totally agree, but that is because my notion of properties is sparse and naturalistic. If you identify properties with predicates (which some weird people seem to), then non-existents can have properties like 'absence' or 'nullity'. |
| 10117 | Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman] |
| Full Idea: The concern of mathematical intuitionists was that the use of certain forms of inference generates, not contradiction, but unjustified assertions. | |
| From: report of Luitzen E.J. Brouwer (Intuitionism and Formalism [1912]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This seems to be the real origin of the verificationist idea in the theory of meaning. It is a hugely revolutionary idea - that ideas are not only ruled out of court by contradiction, but that there are other criteria which should also be met. |
| 12450 | The periodic table not only defines the elements, but also excludes other possible elements [Azzouni] |
| Full Idea: The periodic table not only governs what elements there can be, with their properties, but also explicitly excludes others sorts of elements, because the elements are individuated by the number of discrete protons in their nuclei. | |
| From: Jody Azzouni (Deflating Existential Consequence [2004], Ch.7) | |
| A reaction: It has to be central to the thesis of scientific essentialism that the possibilities in nature are far more restricted than is normally thought, and this observation illustrates the view nicely. He makes a similar point about subatomic particles. |