123 ideas
| 16841 | Good inference has mechanism, precision, scope, simplicity, fertility and background fit [Lipton] |
| 10237 | Coherence is a primitive, intuitive notion, not reduced to something formal [Shapiro] |
| 16854 | Contrary pairs entail contradictions; one member entails negation of the other [Lipton] |
| 10204 | An 'implicit definition' gives a direct description of the relations of an entity [Shapiro] |
| 10206 | Modal operators are usually treated as quantifiers [Shapiro] |
| 10208 | Axiom of Choice: some function has a value for every set in a given set [Shapiro] |
| 10252 | The Axiom of Choice seems to license an infinite amount of choosing [Shapiro] |
| 10207 | Anti-realists reject set theory [Shapiro] |
| 10259 | The two standard explanations of consequence are semantic (in models) and deductive [Shapiro] |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| 10253 | Either logic determines objects, or objects determine logic, or they are separate [Shapiro] |
| 10251 | The law of excluded middle might be seen as a principle of omniscience [Shapiro] |
| 10212 | Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and' [Shapiro] |
| 10209 | A function is just an arbitrary correspondence between collections [Shapiro] |
| 10268 | Maybe plural quantifiers should be understood in terms of classes or sets [Shapiro] |
| 10235 | A sentence is 'satisfiable' if it has a model [Shapiro] |
| 10240 | Model theory deals with relations, reference and extensions [Shapiro] |
| 10239 | The central notion of model theory is the relation of 'satisfaction' [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10234 | Any theory with an infinite model has a model of every infinite cardinality [Shapiro] |
| 10201 | Virtually all of mathematics can be modeled in set theory [Shapiro] |
| 10213 | Real numbers are thought of as either Cauchy sequences or Dedekind cuts [Shapiro] |
| 18243 | Understanding the real-number structure is knowing usage of the axiomatic language of analysis [Shapiro] |
| 18245 | Cuts are made by the smallest upper or largest lower number, some of them not rational [Shapiro] |
| 10236 | There is no grounding for mathematics that is more secure than mathematics [Shapiro] |
| 10256 | For intuitionists, proof is inherently informal [Shapiro] |
| 10202 | Natural numbers just need an initial object, successors, and an induction principle [Shapiro] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 8203 | All the arithmetical entities can be reduced to classes of integers, and hence to sets [Quine] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 10218 | Baseball positions and chess pieces depend entirely on context [Shapiro] |
| 10224 | The even numbers have the natural-number structure, with 6 playing the role of 3 [Shapiro] |
| 10228 | Could infinite structures be apprehended by pattern recognition? [Shapiro] |
| 10230 | The 4-pattern is the structure common to all collections of four objects [Shapiro] |
| 10249 | The main mathematical structures are algebraic, ordered, and topological [Shapiro] |
| 10273 | Some structures are exemplified by both abstract and concrete [Shapiro] |
| 10276 | Mathematical structures are defined by axioms, or in set theory [Shapiro] |
| 10270 | The main versions of structuralism are all definitionally equivalent [Shapiro] |
| 10221 | Is there is no more to structures than the systems that exemplify them? [Shapiro] |
| 10248 | Number statements are generalizations about number sequences, and are bound variables [Shapiro] |
| 10220 | Because one structure exemplifies several systems, a structure is a one-over-many [Shapiro] |
| 10223 | There is no 'structure of all structures', just as there is no set of all sets [Shapiro] |
| 8703 | Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics [Shapiro, by Friend] |
| 10274 | Does someone using small numbers really need to know the infinite structure of arithmetic? [Shapiro] |
| 10200 | We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false) [Shapiro] |
| 10210 | If mathematical objects are accepted, then a number of standard principles will follow [Shapiro] |
| 10215 | Platonists claim we can state the essence of a number without reference to the others [Shapiro] |
| 10233 | Platonism must accept that the Peano Axioms could all be false [Shapiro] |
| 10244 | Intuition is an outright hindrance to five-dimensional geometry [Shapiro] |
| 10280 | A stone is a position in some pattern, and can be viewed as an object, or as a location [Shapiro] |
| 10254 | Can the ideal constructor also destroy objects? [Shapiro] |
| 10255 | Presumably nothing can block a possible dynamic operation? [Shapiro] |
| 10279 | Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules? [Shapiro] |
| 10227 | The abstract/concrete boundary now seems blurred, and would need a defence [Shapiro] |
| 10226 | Mathematicians regard arithmetic as concrete, and group theory as abstract [Shapiro] |
| 10262 | Fictionalism eschews the abstract, but it still needs the possible (without model theory) [Shapiro] |
| 10277 | Structuralism blurs the distinction between mathematical and ordinary objects [Shapiro] |
| 10272 | The notion of 'object' is at least partially structural and mathematical [Shapiro] |
| 10275 | A blurry border is still a border [Shapiro] |
| 10258 | Logical modalities may be acceptable, because they are reducible to satisfaction in models [Shapiro] |
| 10266 | Why does the 'myth' of possible worlds produce correct modal logic? [Shapiro] |
| 16814 | Understanding is not mysterious - it is just more knowledge, of causes [Lipton] |
| 16825 | How do we distinguish negative from irrelevant evidence, if both match the hypothesis? [Lipton] |
| 16851 | The inference to observables and unobservables is almost the same, so why distinguish them? [Lipton] |
| 16799 | Inductive inference is not proof, but weighing evidence and probability [Lipton] |
| 16798 | We infer from evidence by working out what would explain that evidence [Lipton] |
| 16856 | It is more impressive that relativity predicted Mercury's orbit than if it had accommodated it [Lipton] |
| 16857 | Predictions are best for finding explanations, because mere accommodations can be fudged [Lipton] |
| 16827 | If we make a hypothesis about data, then a deduction, where does the hypothesis come from? [Lipton] |
| 16804 | Induction is repetition, instances, deduction, probability or causation [Lipton] |
| 16823 | Standard induction does not allow for vertical inferences, to some unobservable lower level [Lipton] |
| 16800 | An inductive inference is underdetermined, by definition [Lipton] |
| 16858 | We can argue to support our beliefs, so induction will support induction, for believers in induction [Lipton] |
| 16832 | If something in ravens makes them black, it may be essential (definitive of ravens) [Lipton] |
| 16836 | My shoes are not white because they lack some black essence of ravens [Lipton] |
| 16831 | A theory may explain the blackness of a raven, but say nothing about the whiteness of shoes [Lipton] |
| 16833 | We can't turn non-black non-ravens into ravens, to test the theory [Lipton] |
| 16834 | To pick a suitable contrast to ravens, we need a hypothesis about their genes [Lipton] |
| 16802 | Bayes seems to rule out prior evidence, since that has a probability of one [Lipton] |
| 16801 | A hypothesis is confirmed if an unlikely prediction comes true [Lipton] |
| 16837 | Bayes involves 'prior' probabilities, 'likelihood', 'posterior' probability, and 'conditionalising' [Lipton] |
| 16839 | Explanation may be an important part of implementing Bayes's Theorem [Lipton] |
| 16803 | Bayes is too liberal, since any logical consequence of a hypothesis confirms it [Lipton] |
| 16850 | Explanation may describe induction, but may not show how it justifies, or leads to truth [Lipton] |
| 16807 | An explanation gives the reason the phenomenon occurred [Lipton] |
| 16808 | An explanation is what makes the unfamiliar familiar to us [Lipton] |
| 16806 | An explanation is what is added to knowledge to yield understanding [Lipton] |
| 16822 | Seaching for explanations is a good way to discover the structure of the world [Lipton] |
| 16816 | In 'contrastive' explanation there is a fact and a foil - why that fact, rather than this foil? [Lipton] |
| 16826 | With too many causes, find a suitable 'foil' for contrast, and the field narrows right down [Lipton] |
| 16811 | An explanation unifies a phenomenon with our account of other phenomena [Lipton] |
| 16810 | Deduction explanation is too easy; any law at all will imply the facts - together with the facts! [Lipton] |
| 16829 | We reject deductive explanations if they don't explain, not if the deduction is bad [Lipton] |
| 16809 | Good explanations may involve no laws and no deductions [Lipton] |
| 16812 | An explanation shows why it was necessary that the effect occurred [Lipton] |
| 16846 | A cause may not be an explanation [Lipton] |
| 16813 | To explain is to give either the causal history, or the causal mechanism [Lipton] |
| 16815 | Mathematical and philosophical explanations are not causal [Lipton] |
| 16849 | Explanations may be easier to find than causes [Lipton] |
| 16848 | Causal inferences are clearest when we can manipulate things [Lipton] |
| 16842 | We want to know not just the cause, but how the cause operated [Lipton] |
| 16840 | To maximise probability, don't go beyond your data [Lipton] |
| 16824 | Is Inference to the Best Explanation nothing more than inferring the likeliest cause? [Lipton] |
| 16817 | Best Explanation as a guide to inference is preferable to best standard explanations [Lipton] |
| 16818 | The 'likeliest' explanation is the best supported; the 'loveliest' gives the most understanding [Lipton] |
| 16819 | IBE is inferring that the best potential explanation is the actual explanation [Lipton] |
| 16820 | Finding the 'loveliest' potential explanation links truth to understanding [Lipton] |
| 16828 | IBE is not passive treatment of data, but involves feedback between theory and data search [Lipton] |
| 16844 | A contrasting difference is the cause if it offers the best explanation [Lipton] |
| 16853 | We select possible explanations for explanatory reasons, as well as choosing among them [Lipton] |
| 16821 | Must we only have one explanation, and must all the data be made relevant? [Lipton] |
| 16838 | Bayesians say best explanations build up an incoherent overall position [Lipton] |
| 16855 | The best theory is boring: compare 'all planets move elliptically' with 'most of them do' [Lipton] |
| 16852 | Best explanation can't be a guide to truth, because the truth must precede explanation [Lipton] |
| 10203 | We apprehend small, finite mathematical structures by abstraction from patterns [Shapiro] |
| 10229 | Simple types can be apprehended through their tokens, via abstraction [Shapiro] |
| 10217 | We can apprehend structures by focusing on or ignoring features of patterns [Shapiro] |
| 9554 | We can focus on relations between objects (like baseballers), ignoring their other features [Shapiro] |
| 10231 | Abstract objects might come by abstraction over an equivalence class of base entities [Shapiro] |
| 8202 | Meaning is essence divorced from things and wedded to words [Quine] |
| 8201 | The distinction between meaning and further information is as vague as the essence/accident distinction [Quine] |
| 16847 | Counterfactual causation makes causes necessary but not sufficient [Lipton] |