39 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. |
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
| 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. |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
| A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| Full Idea: Ordinal numbers represent order relations. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
| A reaction: Colyvan gives an example, of differentiating a polynomial. |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
| A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
| A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
| 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. |
| 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'. |
| 22868 | The value and truth of knowledge are measured by success in activity [Dewey] |
| Full Idea: What measures knowledge's value, its correctness and truth, is the degree of its availability for conducting to a successful issue the activities of living beings. | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 4:180), quoted by David Hildebrand - Dewey 2 'Critique' | |
| A reaction: Note that this is the measure of truth, not the nature of truth (which James seemed to believe). Dewey gives us a clear and perfect statement of the pragmatic view of knowledge. I don't agree with it. |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
| A reaction: [compressed] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
| A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
| A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
| 22865 | Habits constitute the self [Dewey] |
| Full Idea: All habits are demands for certain kinds of activity; and they constitute the self. | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 14:22), quoted by David Hildebrand - Dewey 1 'Acts' | |
| A reaction: Not an idea I have encountered elsewhere. He emphasises that habits are not repeated actions, but are dispositions. I'm not clear whether these habits must be conscious. |
| 22871 | The good people are those who improve; the bad are those who deteriorate [Dewey] |
| Full Idea: The bad man is the man who no matter how good he has been is beginning to deteriorate, to grow less good. The good man is the man who no matter how morally unworthy he has been is moving to become better. | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 12:181), quoted by David Hildebrand - Dewey 3 'Reconstruct' | |
| A reaction: Although a slightly improving rat doesn't sound as good as a slightly deteriorating saint, I have some sympathy with this thought. The desire to improve seems to be right at the heart of what makes good character. |
| 22876 | Democracy is the development of human nature when it shares in the running of communal activities [Dewey] |
| Full Idea: Democracy is but a name for the fact that human nature is developed only when its elements take part in directing things which are common, things for the sake of which men and women form groups. | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 12:199), quoted by David Hildebrand - Dewey 4 'Democracy' | |
| A reaction: It is hard to prove that human nature develops when it particpates in groups. If people are excluded from power, their loyalty tends to switch to sub-groups, such as friends in a pub, or a football team. Powerless nationalists baffle me. |
| 22875 | Democracy is not just a form of government; it is a mode of shared living [Dewey] |
| Full Idea: A democracy is more than a form of government; it is primarily a mode of associated living, of conjoint communicated experience | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 9:93), quoted by David Hildebrand - Dewey 4 'Democracy' | |
| A reaction: This precisely pinpoints the heart of the culture wars in 2021. A huge swathe of western populations believe in Dewey's idea, but a core of wealthy right-wingers and their servants only see democracy as the mechanism for obtaining power. |
| 22874 | Individuality is only developed within groups [Dewey] |
| Full Idea: Only in social groups does a person have a chance to develop individuality. | |
| From: John Dewey (The Middle Works (15 vols, ed Boydston) [1910], 15:176), quoted by David Hildebrand - Dewey 4 'Individuals' | |
| A reaction: This is a criticism of both Rawls and Nozick. Rawls's initial choosers don't consult, or have much social background. Nozick's property owners ignore everything except contracts. |
| 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. |