32 ideas
| 12463 | Unlike correspondence, truthmaking can be one truth to many truthmakers, or vice versa [Jacobs] |
| Full Idea: I assume a form of truthmaking theory, ..which is a many-many relation, unlike, say correspondence, so that one entity can make multiple truths true and one truth can have multiple truthmakers. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §1) | |
| A reaction: This sounds like common sense, once you think about it. One tree makes many things true, and one statement about trees is made true by many trees. |
| 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. |
| 23476 | Logical constants seem to be entities in propositions, but are actually pure form [Russell] |
| Full Idea: 'Logical constants', which might seem to be entities occurring in logical propositions, are really concerned with pure form, and are not actually constituents of the propositions in the verbal expressions of which their names occur. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], 1.IX) | |
| A reaction: This seems to entirely deny the existence of logical constants, and yet he says that they are named. Russell was obviously under pressure here from Wittgenstein. |
| 23477 | We use logical notions, so they must be objects - but I don't know what they really are [Russell] |
| Full Idea: Such words as or, not, all, some, plainly involve logical notions; since we use these intelligently, we must be acquainted with the logical objects involved. But their isolation is difficult, and I do not know what the logical objects really are. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], 1.IX) | |
| A reaction: See Idea 23476, from the previous page. Russell is struggling. Wittgenstein was telling him that the constants are rules (shown in truth tables), rather than objects. |
| 18273 | Logical truths are known by their extreme generality [Russell] |
| Full Idea: A touchstone by which logical propositions may be distinguished from all others is that they result from a process of generalisation which has been carried to its utmost limits. | |
| From: Bertrand Russell (The Theory of Knowledge [1913], p.129), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 'What' |
| 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. |
| 22315 | There can't be a negative of a complex, which is negated by its non-existence [Potter on Russell] |
| Full Idea: On Russell's pre-war conception it is obvious that a complex cannot be negative. If a complex were true, what would make it false would be its non-existence, not the existence of some other complex. | |
| From: comment on Bertrand Russell (The Theory of Knowledge [1913]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 41 'Neg' | |
| A reaction: It might be false because it doesn't exist, but also 'made' false by a rival complex (such as Desdemona loving Othello). |
| 14375 | If structures result from intrinsic natures of properties, the 'relations' between them can drop out [Jacobs] |
| Full Idea: If a relation holds between two properties as a result of their intrinsic natures, then it appears the relation between the properties is not needed to do the structuring of reality; the properties themselves suffice to fix the structure. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.1) | |
| A reaction: [the first bit quotes Jubien 2007] He cites a group of scientific essentialists as spokesmen for this view. Sounds right to me. No on seems able to pin down what a relation is - which may be because there is no such entity. |
| 14378 | Science aims at identifying the structure and nature of the powers that exist [Jacobs] |
| Full Idea: Scientific practice seems aimed precisely at identifying the structure and nature of the powers that exist. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.3) | |
| A reaction: Good. Friends of powers should look at this nice paper by Jacobs. There is a good degree of support for this view from pronouncements of modern scientists. If scientists don't support it, they should. Otherwise they are trapped in the superficial. |
| 12467 | Powers come from concrete particulars, not from the laws of nature [Jacobs] |
| Full Idea: The source of powers is not the laws of nature; it is the powerful nature of the ordinary properties of concrete particulars. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2) | |
| A reaction: This pithily summarises my own view. People who think the powers of the world derive from the laws either have an implicit religious framework, or they are giving no thought at all to the ontological status of the laws. |
| 14377 | Possibilities are manifestations of some power, and impossibilies rest on no powers [Jacobs] |
| Full Idea: To be possible is just to be one of the many manifestations of some power, and to be impossible is to be a manifestation of no power. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2.1) | |
| A reaction: [This remark occurs in a discussion of theistic Aristotelianism] I like this. If we say that something is possible, the correct question is to ask what power could bring it about. |
| 14376 | States of affairs are only possible if some substance could initiate a causal chain to get there [Jacobs] |
| Full Idea: A non-actual state of affairs in possible if there actually was a substance capable of initiating a causal chain, perhaps non-deterministic, that could lead to the state of affairs that we claim is possible. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.2) | |
| A reaction: [He is quoting A.R. Pruss 2002] That seems exactly right. Of course the initial substance(s) might create a further substance, such as a transuranic element, which then produces the state of affairs. I favour this strongly actualist view. |
| 14379 | Counterfactuals invite us to consider the powers picked out by the antecedent [Jacobs] |
| Full Idea: A counterfactual is an invitation to consider what the properties picked out by the antecedent are powers for (where Lewis 1973 took it to be an invitation to consider what goes on in a selected possible world). | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.4.3) | |
| A reaction: A beautifully simple proposal from Jacobs, with which I agree. This seems to be an expansion of the Ramsey test for conditionals, where you consider the antecedent being true, and see what follows. What, we ask Ramsey, would make it follow? |
| 14372 | Possible worlds are just not suitable truthmakers for modality [Jacobs] |
| Full Idea: Possible worlds are just not the sorts of things that could ground modality; they are not suitable truthmakers. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: Are possible world theorists actually claiming that the worlds 'ground' modality? Maybe Lewis is, since all those concrete worlds had better do some hard work, but for the ersatzist they just provide a kind of formal semantics, leaving ontology to others. |
| 12466 | All modality is in the properties and relations of the actual world [Jacobs] |
| Full Idea: Properties and the relations between them introduce modal connections in the actual world. ..This is a strong form of actualism, since all of modality is part of the fundamental fabric of the actual world. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4) | |
| A reaction: This is the view of modality which I find most congenial, with the notion of 'powers' giving us the conceptual framework on which to build an account. |
| 14371 | We can base counterfactuals on powers, not possible worlds, and hence define necessity [Jacobs] |
| Full Idea: Together with a definition of possibility and necessity in terms of counterfactuals, the powers semantics of counterfactuals generates a semantics for modality that appeals to causal powers and not possible worlds. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §1) | |
| A reaction: Wonderful. Just what the doctor ordered. The only caveat is that if we say that reality is built up from fundamental powers, then might those powers change their character without losing their identity (e.g. gravity getting weaker)? |
| 12465 | Concrete worlds, unlike fictions, at least offer evidence of how the actual world could be [Jacobs] |
| Full Idea: Lewis's concrete worlds give a better account of modality (than fictional worlds). When I learn that a man like me drives a truck, I gain evidence for the fact that I can drive a truck. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: Cf. Idea 12464. Jacobs still rightly rejects this as an account of possibility, since the possibility that I might drive a truck must be rooted in me, not in some other person who drives a truck, even if that person is very like me. |
| 12464 | If some book described a possibe life for you, that isn't what makes such a life possible [Jacobs] |
| Full Idea: Suppose somewhere deep in the rain forest is a book that includes a story about you as a truck-driver. I doubt that you would be inclined the think that that story, that book, is the reason you could have been a truck driver. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §3) | |
| A reaction: This begins to look like a totally overwhelming and obvious reason why possible worlds (especially as stories) don't give a good metaphysical account of possibility. They provide a semantic structure for modal reasoning, but that is entirely different. |
| 12469 | Possible worlds semantics gives little insight into modality [Jacobs] |
| Full Idea: If we want our semantics for modality to give us insight into the truthmakers for modality, then possible worlds semantics is inadequate. | |
| From: Jonathan D. Jacobs (A Powers Theory of Modality [2010], §4.4) | |
| A reaction: [See the other ideas of Jacobs (and Jubien) for this] It is an interesting question whether a semantics for a logic is meant to give us insight into how things really are, or whether it just builds nice models. Satisfaction, or truth? |
| 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. |