40 ideas
| 11159 | My account shows how the concept works, rather than giving an analysis [Fine,K] |
| Full Idea: My assimilation of essence to definition ...may not provide us with an analysis of the concept, but it does provide us with a good model of how the concept works. | |
| From: Kit Fine (Essence and Modality [1994], p. 3) | |
| A reaction: An example of the modern shift in analytic philosophy, away from the dream of given a complete analysis of a concept, towards giving an account of the concepts relationships. Compare Shoemaker in Idea 8559. |
| 11157 | Modern philosophy has largely abandoned real definitions, apart from sortals [Fine,K] |
| Full Idea: In modern analytic philosophy we find that, as a result of sustained empiricist critique, the idea of real definition has been more or less given up (unless it be taken to be vestigially present in the notion of a sortal). | |
| From: Kit Fine (Essence and Modality [1994], p. 3) | |
| A reaction: The account of essences as falling under sortals (roughly, categorising terms) is associated with David Wiggins. Kit Fine is in the business of reviving Aristotelian real definitions, as are fans of scientific essentialism (see under 'Nature'). |
| 11171 | Defining a term and giving the essence of an object don't just resemble - they are the same [Fine,K] |
| Full Idea: There is an analogy between defining a term and giving the essence of an object. ..However, I am inclined to think that the two cases are not merely parallel but are, at bottom, the same. | |
| From: Kit Fine (Essence and Modality [1994], p.13) | |
| A reaction: The proposal is something like the meaning of a concept being the essence of the concept. And essence is definition. The parallel is that they both lead to necessities, either derived from objects or from concepts. Sounds good to me. |
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 11151 | An object is dependent if its essence prevents it from existing without some other object [Fine,K] |
| Full Idea: One object depends upon another (in one sense of the term) if its essence prevents it from existing without the other object. | |
| From: Kit Fine (Essence and Modality [1994], p. 2) | |
| A reaction: I take the interest of this to be that essences are usually thought to be intrinsic, but this seems to involve the object in necessary external relations. |
| 11152 | Essences are either taken as real definitions, or as necessary properties [Fine,K] |
| Full Idea: Essence has been conceived either on the model of definition, involving the 'real' as opposed to 'nominal' definitions, or it is elucidated in modal terms, located in de re cases of modal attributions (an object being necessarily a certain way). | |
| From: Kit Fine (Essence and Modality [1994], p. 2) | |
| A reaction: [compressed] Fine sets out to defend the definitional view, which derives from Aristotle, his line being that necessity depends on essence, and so cannot be used to define it. I think I agree. |
| 11161 | Essentially having a property is naturally expressed as 'the property it must have to be what it is' [Fine,K] |
| Full Idea: We have an informal way of saying an object essentially has a property, as 'the object must have the property if it is to be the object that it is', and this form of words manages to convey what we wish to convey. | |
| From: Kit Fine (Essence and Modality [1994], p. 4) | |
| A reaction: The importance of this claim is that it makes no mention of 'necessity'. Fine's view is plausible, but hard to evaluate once he has said. We seem to then divide an object's properties into identity properties, causal properties and peripheral properties. |
| 11160 | Simple modal essentialism refers to necessary properties of an object [Fine,K] |
| Full Idea: The simplest form of the modal account takes an object to have a property essentially just in case it is necessary that the object has the property. | |
| From: Kit Fine (Essence and Modality [1994], p. 3) | |
| A reaction: Fine wants to reverse the account, explaining necessities in terms of prior essences. |
| 11158 | Essentialist claims can be formulated more clearly with quantified modal logic [Fine,K] |
| Full Idea: With the advent of quantified modal logic, philosophers have been in a better position to formulate essentialist claims. | |
| From: Kit Fine (Essence and Modality [1994], p. 3) | |
| A reaction: A nice illustration of the role which logic plays in modern analytic philosophy. It is not an unreasonable assumption that we will understand a theoretical problem more clearly if we can articulate it more accurately. |
| 11167 | Metaphysical necessity is a special case of essence, not vice versa [Fine,K] |
| Full Idea: Far from viewing essence as a special case of metaphysical necessity, we should view metaphysical necessity as a special case of essence. | |
| From: Kit Fine (Essence and Modality [1994], p. 9) | |
| A reaction: This strikes me as one of the most powerful proposals in modern philosophy (even if it is a reiteration of Aristotle!). |
| 16537 | Essence as necessary properties produces a profusion of essential properties [Fine,K, by Lowe] |
| Full Idea: If an essence is a sum of essential properties (had in all possible worlds where it exists), Fine points out that it seems grossly to overgenerate essential properties ('S is either a man or a mouse', or 'S is such that 2+2=4'). | |
| From: report of Kit Fine (Essence and Modality [1994]) by E.J. Lowe - What is the Source of Knowledge of Modal Truths? 6 | |
| A reaction: To me this is the sort of mess you get into when you accept that 'being such that p' is a property. Defenders of the modal approach always have to eliminate 'trivial' properties from essences, but non-trivial is a defining feature of an essence. |
| 11163 | The nature of singleton Socrates has him as a member, but not vice versa [Fine,K] |
| Full Idea: Can we not recognise a sense of 'what an object is', according to which it lies in the nature of a singleton to have Socrates as a member, even though it does not lie in the nature of Socrates to belong to the singleton? | |
| From: Kit Fine (Essence and Modality [1994], p. 5) | |
| A reaction: Important and persuasive. It echoes the example in Idea 11162, that the necessary relation is not part of the essence. Socrates is necessarily in {Socrates}, but that is because of the set, not because of Socrates. Essences causes necessities. |
| 11164 | It is not part of the essence of Socrates that a huge array of necessary truths should hold [Fine,K] |
| Full Idea: Necessarily any necessary truth will hold if Socrates exists. But it is no part of Socrates' essence that there be infinitely many prime numbers, ..or that objects like the Eiffel Tower have their own necessary essence. | |
| From: Kit Fine (Essence and Modality [1994], p. 5-6) | |
| A reaction: This and the 'singleton Socrates' example (Idea 11165) are the twin prongs of Fine's attack on the modal account of essentialism. I think they constitute one of the best single pages in the whole of recent philosophy. Bravo. |
| 10935 | An essential property of something must be bound up with what it is to be that thing [Fine,K, by Rami] |
| Full Idea: Fine's view is that the notion of an essential property of a thing should be bound up with the notion of what it is to be that thing (unlike, for example, Socrates being such that there are infinitely many primes). | |
| From: report of Kit Fine (Essence and Modality [1994]) by Adolph Rami - Essential vs Accidental Properties §2 | |
| A reaction: I would think that Fine is so obviously right that it was hardly worth saying, but philosophers are a funny lot, and are quite likely to claim that features of prime numbers are part of the essence of a long-dead philosopher. |
| 10936 | Essential properties are part of an object's 'definition' [Fine,K, by Rami] |
| Full Idea: According to Fine's definitional characterization of essential properties, they are those of an object's properties that are part of the object's 'definition'. | |
| From: report of Kit Fine (Essence and Modality [1994]) by Adolph Rami - Essential vs Accidental Properties §2 | |
| A reaction: This demands not just an account of what a definition is, but also the notion that there is only one fixed and correct definition (since the object presumably only has one essence) - but there seems to be something relative about a good definition. |
| 11165 | If Socrates lacks necessary existence, then his nature cannot require his parents' existence [Fine,K] |
| Full Idea: If there is nothing in the nature of Socrates which demands that he exists, then presumably there is nothing in the nature of Socrates which demands that his parents exist. | |
| From: Kit Fine (Essence and Modality [1994], p. 6) | |
| A reaction: This sounds conclusive to me, against any claim that Socrates necessarily had those parents, if the claim is based on the identity or esssence of Socrates. |
| 11166 | The subject of a proposition need not be the source of its necessity [Fine,K] |
| Full Idea: We naturally suppose, if a subject-predicate proposition is necessary, that the subject of the proposition is the source of the necessity. But that singleton 2 contains 2 is necessary, whether the number or the set is the subject of the proposition. | |
| From: Kit Fine (Essence and Modality [1994], p. 9) | |
| A reaction: A very nice addition to his general attack on the idea that essence should be accounted for in terms of his necessity. He asks a beautifully simple question: for each necessity that we accept, what is the source of that necessity? |
| 11169 | Conceptual necessities rest on the nature of all concepts [Fine,K] |
| Full Idea: Conceptual (and logical) necessities can be taken to be the propositions which are true in virtue of the nature of all concepts (or just the logical concepts). | |
| From: Kit Fine (Essence and Modality [1994], p. 9-10) | |
| A reaction: The idea that something might be true simply because of the nature of a concept sounds good, and a slightly better formulation than traditional accounts of analytic truth. |
| 11162 | Socrates is necessarily distinct from the Eiffel Tower, but that is not part of his essence [Fine,K] |
| Full Idea: It is necessary that Socrates and the Eiffel Tower be distinct. But it is not essential to Socrates that he be distinct from the Tower, for there is nothing in his nature which connects him in any special way to it. | |
| From: Kit Fine (Essence and Modality [1994], p. 5) | |
| A reaction: I find this simple argument very persuasive in separating out necessary facts about an object from the essence of that object. |
| 11168 | Metaphysical necessities are true in virtue of the nature of all objects [Fine,K] |
| Full Idea: The metaphysically necessary truths can be identified with the propositions which are true in virtue of the nature of all objects whatever. | |
| From: Kit Fine (Essence and Modality [1994], p. 9) | |
| A reaction: This is part of Fine's proposal that necessities are derived from the essences or natures of things, which view I find very congenial. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 11170 | Analytic truth may only be true in virtue of the meanings of certain terms [Fine,K] |
| Full Idea: Just as a necessary truth may be true in virtue of the identity of certain objects as opposed to others, so an analytic truth may be true in virtue of the meanings of certain terms as opposed to others (such as 'bachelor' rather than 'unmarried'). | |
| From: Kit Fine (Essence and Modality [1994], p.10) | |
| A reaction: This is a beautifully simple observation, that the necessity of 'bachelors are unmarried men' derives from part of the proposition, not from the whole of it. So what is it about the part that generates the apparent necessity? The nature of the concept! |
| 11172 | The meaning of 'bachelor' is irrelevant to the meaning of 'unmarried man' [Fine,K] |
| Full Idea: Strictly speaking it is irrelevant to the meaning of 'bachelor' that the phrase 'unmarried man' means what it does. | |
| From: Kit Fine (Essence and Modality [1994], p.13) | |
| A reaction: His point is that the necessary truth here derives from the meaning of 'bachelor', and not from the meaning of 'unmarried man'. But is also true that 'unmarried man' means 'bachelor' (for those familiar with the latter, but not the former). |
| 23674 | If an attempted poisoning results in benefits, we still judge the agent a poisoner [Reid] |
| Full Idea: If a man should give to his neighbour a potion which he really believes will poison him, but which, in the event, proves salutary, and does much good; in moral estimation, he is a poisoner, and not a benefactor. | |
| From: Thomas Reid (Essays on Active Powers 3: Princs of action [1788], 5) | |
| A reaction: I take Reid to mean that morality concerns how we assess the agent, and not the results of his actions. Mill and Bentham concede that we judge people this way, but don't think morality mainly concerns judging people. |
| 23675 | We shouldn't do to others what would be a wrong to us in similar circumstances [Reid] |
| Full Idea: It is a first principle of morals, that we ought not to do to another what we should think wrong to be done to us in like circumstances. | |
| From: Thomas Reid (Essays on Active Powers 3: Princs of action [1788], 6) | |
| A reaction: This negative form of the rule is more plausible than the positive form, presumably because there is more consensus about what we all dislike than what we all prefer. But presents for people that they would like, not that you like. |
| 23672 | To be virtuous, we must care about duty [Reid] |
| Full Idea: A man cannot be virtuous, if he has no regard to duty. | |
| From: Thomas Reid (Essays on Active Powers 3: Princs of action [1788], 5) | |
| A reaction: Thus are Aristotle and Kant united in a simple sentence. Aristotle thinks that a virtuous person thereby sees what is the right thing to do, but I take 'duty' to imply a requirement which comes not from good character but from external society. |
| 23673 | Every worthy man has a principle of honour, and knows what is honourable [Reid] |
| Full Idea: I presume it will be granted, that, in every man of real worth, there is a principle of honour, a regard to what is honourable or dishonourable, very distinct from a regard to his interest. | |
| From: Thomas Reid (Essays on Active Powers 3: Princs of action [1788], 5) | |
| A reaction: Note that there is a 'principle' of honour in a person's character, and there are also actions which are intrinsically honourable or not. I fear that only the worthy are honourable, and only the honourable are worthy! |