83 ideas
| 1815 | Reasoning needs arbitrary faith in preliminary hypotheses (Mode 14) [Agrippa, by Diog. Laertius] |
| 1812 | All discussion is full of uncertainty and contradiction (Mode 11) [Agrippa, by Diog. Laertius] |
| 1811 | Proofs often presuppose the thing to be proved (Mode 15) [Agrippa, by Diog. Laertius] |
| 1813 | All reasoning endlessly leads to further reasoning (Mode 12) [Agrippa, by Diog. Laertius] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 22246 | A train of reasoning must be treated as all happening simultaneously [Recanati] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 16357 | Mental files are the counterparts of singular terms [Recanati] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 16360 | Identity statements are informative if they link separate mental files [Recanati] |
| 16374 | There is a continuum from acquaintance to description in knowledge, depending on the link [Recanati] |
| 8850 | Agrippa's Trilemma: justification is infinite, or ends arbitrarily, or is circular [Agrippa, by Williams,M] |
| 1814 | Everything is perceived in relation to another thing (Mode 13) [Agrippa, by Diog. Laertius] |
| 22248 | How can we communicate indexical thoughts to people not in the right context? [Recanati] |
| 16354 | Indexicality is closely related to singularity, exploiting our direct relations with things [Recanati] |
| 18409 | Indexicals apply to singular thought, and mental files have essentially indexical features [Recanati] |
| 22247 | Indexicality is not just a feature of language; examples show it also occurs in thought [Recanati] |
| 16361 | A mental file treats all of its contents as concerning one object [Recanati] |
| 22242 | Mental files are concepts, which are either collections or (better) containers [Recanati] |
| 22243 | The Frege case of believing a thing is both F and not-F is explained by separate mental files [Recanati] |
| 16370 | A file has a 'nucleus' through its relation to the object, and a 'periphery' of links to other files [Recanati] |
| 16371 | Files can be confused, if two files correctly have a single name, or one file has two names [Recanati] |
| 16373 | Encylopedic files have further epistemic links, beyond the basic one [Recanati] |
| 16375 | Singular thoughts need a mental file, and an acquaintance relation from file to object [Recanati] |
| 16377 | Expected acquaintance can create a thought-vehicle file, but without singular content [Recanati] |
| 16378 | An 'indexed' file marks a file which simulates the mental file of some other person [Recanati] |
| 16387 | Reference by mental files is Millian, in emphasising acquaintance, rather than satisfaction [Recanati] |
| 16358 | The reference of a file is fixed by what it relates to, not the information it contains [Recanati] |
| 16367 | There are transient 'demonstrative' files, habitual 'recognitional' files, cumulative 'encyclopedic' files [Recanati] |
| 16368 | Files are hierarchical: proto-files, then first-order, then higher-order encyclopedic [Recanati] |
| 16381 | The content of thought is what is required to understand it (which involves hearers) [Recanati] |
| 16365 | Mental files are individual concepts (thought constituents) [Recanati] |
| 16356 | There may be two types of reference in language and thought: descriptive and direct [Recanati] |
| 16386 | Direct reference is strong Millian (just a tag) or weak Kaplanian (allowing descriptions as well) [Recanati] |
| 16393 | In super-direct reference, the referent serves as its own vehicle of reference [Recanati] |
| 16372 | Sense determines reference says same sense/same reference; new reference means new sense [Recanati] |
| 16388 | We need sense as well as reference, but in a non-descriptive form, and mental files do that [Recanati] |
| 16359 | Sense is a mental file (not its contents); similar files for Cicero and Tully are two senses [Recanati] |
| 16355 | Problems with descriptivism are reference by perception, by communications and by indexicals [Recanati] |
| 16348 | Descriptivism says we mentally relate to objects through their properties [Recanati] |
| 16384 | Definite descriptions reveal either a predicate (attributive use) or the file it belongs in (referential) [Recanati] |
| 16352 | A rigid definite description can be attributive, not referential: 'the actual F, whoever he is….' [Recanati] |
| 22245 | A linguistic expression refers to what its associated mental file refers to [Recanati] |
| 16353 | Singularity cannot be described, and it needs actual world relations [Recanati] |
| 16382 | Fregean modes of presentation can be understood as mental files [Recanati] |
| 16389 | If two people think 'I am tired', they think the same thing, and they think different things [Recanati] |
| 16363 | Indexicals (like mental files) determine their reference relationally, not by satisfaction [Recanati] |
| 16364 | Indexical don't refer; only their tokens do [Recanati] |
| 16351 | In 2-D semantics, reference is determined, then singularity by the truth of a predication [Recanati] |
| 16350 | Two-D semantics is said to help descriptivism of reference deal with singular objects [Recanati] |
| 16380 | Russellian propositions are better than Fregean thoughts, by being constant through communication [Recanati] |
| 22250 | There are speakers' thoughts and hearers' thoughts, but no further thought attached to the utterance [Recanati] |
| 22249 | The Naive view of communication is that hearers acquire exactly the thoughts of the speaker [Recanati] |