14 ideas
10882 | Predicative definitions only refer to entities outside the defined collection [Horsten] |
Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §2.4) |
18755 | Validity is explained as truth in all models, because that relies on the logical terms [McGee] |
Full Idea: A model of a language assigns values to non-logical terms. If a sentence is true in every model, its truth doesn't depend on those non-logical terms. Hence the validity of an argument comes from its logical form. Thus models explain logical validity. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: [compressed] Thus you get a rigorous account of logical validity by only allowing the rigorous input of model theory. This is the modern strategy of analytic philosophy. But is 'it's red so it's coloured' logically valid? |
18751 | Natural language includes connectives like 'because' which are not truth-functional [McGee] |
Full Idea: Natural language includes connectives that are not truth-functional. In order for 'p because q' to be true, both p and q have to be true, but knowing the simpler sentences are true doesn't determine whether the larger sentence is true. | |
From: Vann McGee (Logical Consequence [2014], 2) |
18761 | Second-order variables need to range over more than collections of first-order objects [McGee] |
Full Idea: To get any advantage from moving to second-order logic, we need to assign to second-order variables a role different from merely ranging over collections made up of things the first-order variables range over. | |
From: Vann McGee (Logical Consequence [2014], 7) | |
A reaction: Thus it is exciting if they range over genuine properties, but not so exciting if you merely characterise those properties as sets of first-order objects. This idea leads into a discussion of plural quantification. |
18753 | An ontologically secure semantics for predicate calculus relies on sets [McGee] |
Full Idea: We can get a less ontologically perilous presentation of the semantics of the predicate calculus by using sets instead of concepts. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: The perilous versions rely on Fregean concepts, and notably Russell's 'concept that does not fall under itself'. The sets, of course, have to be ontologically secure, and so will involve the iterative conception, rather than naive set theory. |
18754 | Logically valid sentences are analytic truths which are just true because of their logical words [McGee] |
Full Idea: Logically valid sentences are a species of analytic sentence, being true not just in virtue of the meanings of their words, but true in virtue of the meanings of their logical words. | |
From: Vann McGee (Logical Consequence [2014], 4) | |
A reaction: A helpful link between logical truths and analytic truths, which had not struck me before. |
10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §5.2) |
18757 | Soundness theorems are uninformative, because they rely on soundness in their proofs [McGee] |
Full Idea: Soundness theorems are seldom very informative, since typically we use informally, in proving the theorem, the very same rules whose soundness we are attempting to establish. | |
From: Vann McGee (Logical Consequence [2014], 5) | |
A reaction: [He cites Quine 1935] |
10885 | Computer proofs don't provide explanations [Horsten] |
Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §5.3) |
18760 | The culmination of Euclidean geometry was axioms that made all models isomorphic [McGee] |
Full Idea: One of the culminating achievements of Euclidean geometry was categorical axiomatisations, that describe the geometric structure so completely that any two models of the axioms are isomorphic. The axioms are second-order. | |
From: Vann McGee (Logical Consequence [2014], 7) | |
A reaction: [He cites Veblen 1904 and Hilbert 1903] For most mathematicians, categorical axiomatisation is the best you can ever dream of (rather than a single true axiomatisation). |
10881 | The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten] |
Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept. | |
From: Leon Horsten (Philosophy of Mathematics [2007], §2.3) |
18762 | A maxim claims that if we are allowed to assert a sentence, that means it must be true [McGee] |
Full Idea: If our linguistic conventions entitle us to assert a sentence, they thereby make it true, because of the maxim that 'truth is the norm of assertion'. | |
From: Vann McGee (Logical Consequence [2014], 8) | |
A reaction: You could only really deny that maxim if you had no belief at all in truth, but then you can assert anything you like (with full entitlement). Maybe you can assert anything you like as long as it doesn't upset anyone? Etc. |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |