38 ideas
| 16539 | A definition of a circle will show what it is, and show its generating principle [Lowe] |
| Full Idea: If the definition of a circle is based on 'locus of a point', this tells us what a circle is, and it does so by revealing its generating principle, what it takes for a circle to come into being. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: Lowe says that real definitions, as essences, do not always have to spell out a 'generating principle', but they do in this case. Another approach would be to try to map dependence relations between truths about circles, and see what is basic. |
| 16540 | Defining an ellipse by conic sections reveals necessities, but not the essence of an ellipse [Lowe] |
| Full Idea: Defining an ellipse in terms of the oblique intersection of a cone and a plane (rather than in terms of the sum of the distance between the foci) gives us a necessary property, but not the essence, because the terms are extrinsic to its nature. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: [compressed wording] Helpful and illuminating. If you say some figure is what results when one thing intersects another, that doesn't tell you what the result actually is. Geometrical essences may be a bit vague, but they are quite meaningful. |
| 16548 | An essence is what an entity is, revealed by a real definition; this is not an entity in its own right [Lowe] |
| Full Idea: An entity's essence is just what that entity is, revealed by its real definition. This isn't a distinct entity, but either the entity itself, or (my view) no entity at all. ..We should not reify essence, as that leads to an infinite regress of essences. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: The regress problem is a real one, if we wish to treat an essence as some proper and distinct part of an entity. If it is a mechanism, for example, the presumably a mechanism has an essence. No, it doesn't! Levels of explanation! |
| 16549 | Simple things like 'red' can be given real ostensive definitions [Lowe] |
| Full Idea: Is it true that we cannot say, non-circularly, what red is? We cannot find a complex synonym for it, but I think we can provide red with an ostensive real definition. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: I'm not quite sure how 'real' this definition would be, if it depends on observers (some of whom may be colourblind). In what sense is this act of ostensions a 'definition'? You must distinguish the colour from the texture or shape. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 16545 | The essence of lumps and statues shows that two objects coincide but are numerically distinct [Lowe] |
| Full Idea: It is a metaphysically necessary truth, obtaining in virtue of the essences of such objects (of what a bronze statue and a lump of bronze are) that when it exists a bronze statue coincides with a lump of bronze, which is numerically distinct from it. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: I think it is nonsense to treat the lump and statue as two objects. It is essential that a statue be made of a lump, and essential that a lump have a shape, so to treat the lump and the shape as two different objects is a failure to grasp the essence. |
| 16546 | The essence of a bronze statue shows that it could be made of different bronze [Lowe] |
| Full Idea: It is a metaphysical possibility, obtaining in virtue of the essences of such objects, that the same bronze statue should coincide with different lumps of bronze at different times. (..they have different persistence conditions). | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: If the fame of a statue were that it had been made by melting down the shield of Achilles (say), then the bronze it was made of would be its most important feature. Essences are more contextual than Lowe might wish. |
| 16551 | Grasping an essence is just grasping a real definition [Lowe] |
| Full Idea: All that grasping an essence amounts to is understanding a real definition, that is, understanding a special kind of proposition. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 7) | |
| A reaction: He refuses to 'reify' an essence, and says it is not an entity, so he seems to think that the definition is the essence, but Aristotle and I take the essence to be what is picked out by the correct definition - not the definition itself. |
| 16542 | Explanation can't give an account of essence, because it is too multi-faceted [Lowe] |
| Full Idea: Explanation is a multifaceted one, with many species (logical, mathematical, causal, teleological, and psychological), ..so it is not a notion fit to be appealed to in order to frame a perspicuous account of essence. That is one species of explanation. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: This directly attacks the core of my thesis! His parenthetical list does not give types of explanation. If I say this explanation is 'psychological', that says nothing about what explanation is. All of his instances could rest on essences. |
| 16552 | If we must know some entity to know an essence, we lack a faculty to do that [Lowe] |
| Full Idea: If knowledge of essence were by acquaintance of a special kind of entity, we would doubt our ability to grasp the essence of things. For what faculty could be involved in this special kind of acquaintance? | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 7) | |
| A reaction: This is Lockean empirical scepticism about essences, but I take the view that sometimes you can be acquainted with an essence, but more often you correctly infer it from you acquaintance - and this is just what scientists do. |
| 16533 | Logical necessities, based on laws of logic, are a proper sub-class of metaphysical necessities [Lowe] |
| Full Idea: If logically necessary truths are consequences of the laws of logic, then I think they are only a proper sub-class of the class of metaphysically necessary truths. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 1) | |
| A reaction: The problem for this is unusual and bizarre systems of logic, or systems that contradict one another. This idea is only plausible if you talk about the truths derived from some roughly 'classical' core of logic. 'Tonk' won't do it! |
| 16531 | 'Metaphysical' necessity is absolute and objective - the strongest kind of necessity [Lowe] |
| Full Idea: By 'metaphysical' necessity I mean necessity of the strongest possible kind - absolute necessity - and I take it to be an objective kind of necessity, rather than being something mind-dependent. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 1) | |
| A reaction: See Bob Hale for the possibility that 'absolute' and 'metaphysical' necessity might come apart. I think I believe in metaphysical necessity, but I'm uneasy about 'absolute' necessity. That may be discredited by the sceptics. |
| 16532 | 'Epistemic' necessity is better called 'certainty' [Lowe] |
| Full Idea: 'Epistemic' necessity is more properly to be called 'certainty'. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 1) | |
| A reaction: Sounds wrong. Surely I can be totally certain of a contingent truth? |
| 16543 | If an essence implies p, then p is an essential truth, and hence metaphysically necessary [Lowe] |
| Full Idea: If we can truly affirm that it is part of the essence of some entity that p is the case, then p is an essential truth and so a metaphysically necessary truth. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: This feels too quick. He is trying to expound the idea (which I like) that necessity derives from essences, and not vice versa. Is it a metaphysical necessity that there are no moths in my wardrobe, because mothballs have driven them away? Maybe. |
| 16544 | Metaphysical necessity is either an essential truth, or rests on essential truths [Lowe] |
| Full Idea: A metaphysically necessary truth is a truth which is either an essential truth or a truth that obtains in virtue of the essences of two or more distinct things. Hence all metaphysical necessity is grounded in essence. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: Lowe is endeavouring to give an exposition of the approach advocated by Kit Fine. I divide necessities 'because of' things (such as essences) from necessities 'for' things, such as situations or events. |
| 16538 | We could give up possible worlds if we based necessity on essences [Lowe] |
| Full Idea: If we explicate the notion of metaphysical necessity in terms of the notion of essence, rather than vice versa, this may enable us to dispense with the language of possible worlds as a means of explicating modal statements. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: This is the approach I favour, though I am not convinced that the two approaches are in competition, since essentialism gives the driving force for necessity, whereas possible worlds map the logic and semantics of it. |
| 16534 | 'Intuitions' are just unreliable 'hunches'; over centuries intuitions change enormously [Lowe] |
| Full Idea: I suspect that 'intuitions' and 'hunches' are pretty much the same thing, and pretty useless as sources of knowledge. …Things that seemed intuitively true to our forebears a century or two ago often by no means seem intuitively true to us now. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 2) | |
| A reaction: I don't accept this. Intuitions change a lot over the centuries because the reliable knowledge which informs intuitions has also changed a lot. Arguments and evidence may nail individual truths, but coherence must rest on intuition. |
| 16535 | A concept is a way of thinking of things or kinds, whether or not they exist [Lowe] |
| Full Idea: The nearest I can get to a quick definition is to say that a concept is a way of thinking of some thing or kind of things, whether or not a really existent thing or kind of things. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 2) | |
| A reaction: The focus on 'things' seems rather narrow. Are relations things? He makes concepts sound adverbial, so that there is thinking going on, and then we add 'ways' of doing it. Thinking depends on concepts, not concepts on thinking. |
| 16550 | Direct reference doesn't seem to require that thinkers know what it is they are thinking about [Lowe] |
| Full Idea: It may be objected that currently prevailing causal or 'direct' theories of reference precisely deny that a thinker must know what it is the he or she is thinking about in order to be able to think about it. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 7) | |
| A reaction: Lowe says that at least sometimes we have to know that we are thinking about, so this account of reference can't be universally true. My solution is to pull identity and essence apart. You only need identity, not essence, for reference. |
| 20329 | A work of art is an artifact created for the artworld [Dickie] |
| Full Idea: A work of art is an artifact of a kind created to be presented to an artworld public. | |
| From: George Dickie (The New Institutional Theory of Art [1983], p.53) | |
| A reaction: This is the culminating definition in his paper, deriving originally from Danto, and an improvement of his earlier more complex definition. Since this definition amounts to 'this is art if I say it is art', it doesn't seem to reveal much. |
| 16547 | H2O isn't necessary, because different laws of nature might affect how O and H combine [Lowe] |
| Full Idea: It is not metaphysically necessary that water is composed of H2O molecules, because the natural laws governing the chemical behaviour of hydrogen and oxygen atoms could have been significantly different, so they might not have composed that substance. | |
| From: E.J. Lowe (What is the Source of Knowledge of Modal Truths? [2013], 6) | |
| A reaction: I fear this may be incoherent, as science. See Bird on why salt must dissolve in water. There can't (I suspect) be a law which keeps O and H the same, and yet makes them combine differently. |