42 ideas
17070 | Coherence is consilience, simplicity, analogy, and fitting into a web of belief [Smart] |
Full Idea: I shall make use of the admittedly imprecise notions of consilience, simplicity, analogy and fitting into a web of belief, or in short of 'coherence'. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.06) | |
A reaction: Coherence sounds like a family of tests, rather than a single unified concept. I still like coherence, though. |
17072 | We need comprehensiveness, as well as self-coherence [Smart] |
Full Idea: Not mere self-coherence, but comprehensiveness belongs to the notion of coherence. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.07) |
18334 | The truth-maker idea is usually justified by its explanatory power, or intuitive appeal [Rami] |
Full Idea: The two strategies for justifying the truth-maker principle are that it has an explanatory role (for certain philosophical problems and theses), or that it captures the best philosophical intuition of the situation. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 04) | |
A reaction: I would go for 'intuitive', but not in the sense of a pure intuition, but with 'intuitive' as a shorthand for overall coherence. To me the appeal of truth-maker is its place in a naturalistic view of reality. I love explanation, but not here. |
18335 | There are five problems which the truth-maker theory might solve [Rami] |
Full Idea: It is claimed that truth-makers explain universals, or ontological commitment, or commitment to realism, or to the correspondence theory of truth, or to falsify behaviourism or phenomenalism. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 04) | |
A reaction: [compressed] This expands the view that truth-making is based on its explanatory power, rather than on its intuitive correctness. I take the theory to presuppose realism. I don't believe in universals. It marginalises correspondence. Commitment is good! |
18339 | The truth-making relation can be one-to-one, or many-to-many [Rami] |
Full Idea: The truth-making relation can be one-to-one, or many-many. In the latter case, different truths may have the same truth-maker, and one truth may have different truth-makers. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
A reaction: 'There is at least one cat' obviously has many possible truth-makers. Many statements will be made true by the mere existence of a particular cat (such as 'there is an animal in the room' and 'there is a cat in the room'). Many-many wins? |
18333 | Central idea: truths need truthmakers; and possibly all truths have them, and makers entail truths [Rami] |
Full Idea: The main full-blooded truth-maker principle is that x is true iff there is a y that is its truth-maker. This implies the principles that if x is true x has a truth-maker, and the principle that if x has a truth-maker then x is true. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 03) | |
A reaction: [compressed] Rami calls the second principle 'maximalism' and the third principle 'purism'. To reject maximalism is to hold a more restricted version of truth-makers. That is, the claim is that lots of truths have truth-makers. |
18342 | Most theorists say that truth-makers necessitate their truths [Rami] |
Full Idea: Most truth-maker theorists regard the necessitation of a truth by a truth-maker as a necessary condition of truth-making. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 07) | |
A reaction: It seems to me that reality is crammed full of potential truth-makers, but not crammed full of truths. If there is no thinking in the universe, then there are no truths. If that is false, then what sort of weird beast is a 'truth'? |
18340 | It seems best to assume different kinds of truth-maker, such as objects, facts, tropes, or events [Rami] |
Full Idea: Truthmaker anti-monism holds the view that there are truth-makers of different kinds. For example, objects, facts, tropes or events can all be regarded as truthmakers. Objects seem right for existential truths but not others, so anti-monism seems best. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
A reaction: Presumably we need to identify the different types of truth (analytic, synthetic, general, particular...), and only then ask what truth-makers there are for the different types. To presuppose one type of truthmaker would be crazy. |
18341 | Truth-makers seem to be states of affairs (plus optional individuals), or individuals and properties [Rami] |
Full Idea: As truth-makers, some theorists only accept states of affairs, some only accept individuals and states of affairs, and some only accept individuals and particular properties. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 06) | |
A reaction: It seems to me rash to opt for one of these. Truths come in wide-ranging and subtly different types, and the truth-makers probably have a similar range. Any one of these theories will almost certainly quickly succumb to a counterexample. |
18345 | 'Truth supervenes on being' avoids entities as truth-makers for negative truths [Rami] |
Full Idea: The important advantage of 'truth supervenes on being' is that it can be applied to positive and negative contingent truths, without postulating any entities that are responsible for the truth of negative truths. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 09) | |
A reaction: [For this reason, Lewis favours a possible worlds version of the theory] I fear that it solves that problem by making the truth-maker theory so broad-brush that it not longer says very much, apart from committing it to naturalism. |
18346 | 'Truth supervenes on being' only gives necessary (not sufficient) conditions for contingent truths [Rami] |
Full Idea: The thesis that 'truth supervenes on being' (with or without possible worlds) offers only a necessary condition for the truth of contingent propositions, whereas the standard truth-maker theory offers necessary and sufficient conditions. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 09) | |
A reaction: The point, I suppose, is that the change in being might be irrelevant to the proposition in question, so any old change in being will not ensure a change in the truth of the proposition. Again we ask - but what is this truth about? |
18343 | Maybe a truth-maker also works for the entailments of the given truth [Rami] |
Full Idea: The 'entailment principle' for truth-makers says that if x is a truth-maker for y, and y entails z, then x is a truth-maker for z. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 08) | |
A reaction: I think the correct locution is that 'x is a potential truth-maker for z' (should anyone every formulate z, which in most cases they never will, since the entailments of y are probably infinite). Merricks would ask 'but are y and z about the same thing?'. |
18338 | Truth-making is usually internalist, but the correspondence theory is externalist [Rami] |
Full Idea: Most truth-maker theorists are internalists about the truth-maker relation. ...But the correspondence theory makes truth an external relation to some portion of reality. So a truth-maker internalist should not claim to be a narrow correspondence theorist. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
A reaction: [wording rearranged] Like many of Rami's distinctions in this article, this feels simplistic. Sharp distinctions can only be made using sharp vocabulary, and there isn't much of that around in philosophy! |
18337 | Correspondence theories assume that truth is a representation relation [Rami] |
Full Idea: One guiding intuition concerning a correspondence theory of truth says that the relation that accounts for the truth of a truth-bearer is some kind of representation relation. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
A reaction: I unfashionably cling on to some sort of correspondence theory. The paradigm case is of a non-linguistic animal which forms correct or incorrect views about its environment. Truth is a relation, not a property. I see the truth in a bad representation. |
18347 | Deflationist truth is an infinitely disjunctive property [Rami] |
Full Idea: According to the moderate deflationist truth is an infinitely disjunctive property. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 10) | |
A reaction: [He cites Horwich 1998] That is, I presume, that truth is embodied in an infinity of propositions of the form '"p" is true iff p'. |
18350 | Truth-maker theorists should probably reject the converse Barcan formula [Rami] |
Full Idea: There are good reasons for the truth-maker theorist to reject the converse Barcan formula. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], note 16) | |
A reaction: In the text (p.15) Rami cites the inference from 'necessarily everything exists' to 'everything exists necessarily'. [See Williamson 1999] |
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. |
18336 | Internal relations depend either on the existence of the relata, or on their properties [Rami] |
Full Idea: An internal relation is 'existential' if x and y relate in that way whenever they both exist. An internal relation is 'qualitative' if x and y relate in that way whenever they have certain intrinsic properties. | |
From: Adolph Rami (Introduction: Truth and Truth-Making [2009], 05) | |
A reaction: [compressed - Rami likes to write these things in fashionable quasi-algebra, but I have a strong prejudice in this database for expressing ideas in English; call me old-fashioned] The distinction strikes me as simplistic. I would involve dispositions. |
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. |
17073 | I simply reject evidence, if it is totally contrary to my web of belief [Smart] |
Full Idea: The simplest way of fitting the putative observed phenomena of telepathy or clairvoyance into my web of belief is to refuse to take them at face value. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.07-8) | |
A reaction: Love it. It is very disconcerting for the sceptical naturalist to be faced with adamant claims that the paranormal has occurred, but my response is exactly the same as Smart's. I reject the reports, no matter how passionately they are asserted. |
17077 | The height of a flagpole could be fixed by its angle of shadow, but that would be very unusual [Smart] |
Full Idea: You could imagine a person using the angle from a theodolite to decide a suitable spot to cut the height of the flagpole, …but since such circumstances would be very unusual we naturally say the flagpole subtends the angle because of its height. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.14) | |
A reaction: [compressed; he mentions Van Fraassen 1980:132-3 for a similar point] As a response this seems a bit lame, if the direction is fixed by what is 'usual'. I think the key point is that the direction of explanation is one way or the other, not both. |
17078 | Universe expansion explains the red shift, but not vice versa [Smart] |
Full Idea: The theory of the expansion of the universe renders the red shift no longer puzzling, whereas he expansion of the universe is hardly rendered less puzzling by facts about the red shift. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.15) | |
A reaction: The direction of explanation is, I take it, made obvious by the direction of causation, with questions about what is 'puzzling' as mere side-effects. |
17061 | Explanation of a fact is fitting it into a system of beliefs [Smart] |
Full Idea: I want to characterise explanation of some fact as a matter of fitting belief in this fact into a system of beliefs. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.02) | |
A reaction: Sounds good to me. Simple facts slot straight into daily beliefs, and deep obscure facts are explained when we hook them up to things we have already grasped. Quark theory fits into prior physics of forces, properties etc. |
17074 | Explanations are bad by fitting badly with a web of beliefs, or fitting well into a bad web [Smart] |
Full Idea: An explanation may be bad if it fits only into a bad web of belief. It can also be bad if it fits into a (possibly good) web of belief in a bad sort of way. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.09) | |
A reaction: Nice. If you think someone has an absurd web of beliefs, then it counts against some belief (for you) if it fits beautifully into the other person's belief system. Judgement of coherence comes in at different levels. |
17076 | Deducing from laws is one possible way to achieve a coherent explanation [Smart] |
Full Idea: The Hempelian deductive-nomological model of explanation clearly fits in well with the notion of explanation in terms of coherence. One way of fitting a belief into a system is to show that it is deducible from other beliefs. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.13) | |
A reaction: Smart goes on to reject the law-based deductive approach, for familiar reasons, but at least it has something in common with the Smart view of explanation, which is the one I like. |
17071 | An explanation is better if it also explains phenomena from a different field [Smart] |
Full Idea: One explanation will be a better explanation that another if it also explains a set of phenomena from a different field ('consilience'). | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.07) | |
A reaction: This would count as 'unexpected accommodation', rather than prediction. It is a nice addition to Lipton's comparison of mere accommodation versus prediction as criteria. It sounds like a strong criterion for a persuasive explanation. |
17062 | If scientific explanation is causal, that rules out mathematical explanation [Smart] |
Full Idea: I class mathematical explanation with scientific explanation. This would be resisted by those who, unlike me, regard the notion of causation as essential to scientific explanation. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.02-3) | |
A reaction: I aim to champion mathematical explanation, in terms of axioms etc., so I am realising that my instinctive attraction to exclusively causal explanation won't do. What explanation needs is a direction of dependence. |
17075 | Scientific explanation tends to reduce things to the unfamiliar (not the familiar) [Smart] |
Full Idea: The history of science suggests that most often explanation is reduction to the unfamiliar. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.11) | |
A reaction: Boyle was keen to reduce things to the familiar, but that was early days for science, and some nasty shocks were coming our way. What would Boyle make of quantum non-locality? |
17063 | Unlike Newton, Einstein's general theory explains the perihelion of Mercury [Smart] |
Full Idea: Newtonian celestial mechanics does not explain the advance of the perihelion of Mercury, while Einstein's general theory of relativity does. | |
From: J.J.C. Smart (Explanation - Opening Address [1990], p.03) | |
A reaction: A perfect example of why explanation is the central concept in science, and probably in all epistemological activity. The desire to know is the desire for an explanation. Once the explanation is obvious, we know. |