30 ideas
| 21544 | It seems that when a proposition is false, something must fail to subsist [Russell] |
| Full Idea: It seems that when a proposition is false, something does not subsist which would subsist if the proposition were true. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.76) | |
| A reaction: This looks to me like a commitment by Russell to the truthmaker principle. The negations of false propositions are made true by some failure of existence in the world. |
| 19053 | Logic would be more natural if negation only referred to predicates [Dummett] |
| Full Idea: A better proposal for a formal logic closer to natural language would be one that had a negation-operator only for (simple) predicates. | |
| From: Michael Dummett (Presupposition [1960], p.27) | |
| A reaction: Dummett observes that classical formal logic was never intended to be close to natural language. Term logic does have that aim, but the meta-question is whether that end is desirable, and why. |
| 21539 | Excluded middle can be stated psychologically, as denial of p implies assertion of not-p [Russell] |
| Full Idea: The law of excluded middle may be stated in the form: If p is denied, not-p must be asserted; this form is too psychological to be ultimate, but the point is that it is significant and not a mere tautology. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.41) | |
| A reaction: 'Psychology' is, of course, taboo, post-Frege, though I think it is interesting. Stated in this form the law looks more false than usual. I can be quite clear than p is unacceptable, but unclear about its contrary. |
| 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. |
| 19052 | Natural language 'not' doesn't apply to sentences [Dummett] |
| Full Idea: Natural language does not possess a sentential negation-operator. | |
| From: Michael Dummett (Presupposition [1960], p.27) | |
| A reaction: This is a criticism of Strawson, who criticises logic for not following natural language, but does it himself with negation. In the question of how language and logic connect, this idea seems important. Term Logic aims to get closer to natural language. |
| 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. |
| 21538 | If two people perceive the same object, the object of perception can't be in the mind [Russell] |
| Full Idea: If two people can perceive the same object, as the possibility of any common world requires, then the object of an external perception is not in the mind of the percipient. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.33) | |
| A reaction: This is merely an assertion of the realist view, rather than an argument. I take representative realism to tell a perfectly good story that permits two subjective representations of the same object. |
| 21534 | The only thing we can say about relations is that they relate [Russell] |
| Full Idea: It may be doubted whether relations can be adequately characterised by anything except the fact that they relate. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.27) | |
| A reaction: We can characterise a rope that ties things together. If I say 'stand to his left', do I assume the existence of one of the relata and the relation, but without the second relata? How about 'you two stand over there, with him on the left'? |
| 21540 | Relational propositions seem to be 'about' their terms, rather than about the relation [Russell] |
| Full Idea: In some sense which it would be very desirable to define, a relational proposition seems to be 'about' its terms, in a way in which it is not about the relation. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.53) | |
| A reaction: Identifying how best to specify what a proposition is actually 'about' is a very illuminating mode of enquiry. You can't define 'underneath' without invoking a pair of objects to illustrate it. A proposition can still focus on the relation. |
| 21536 | When I perceive a melody, I do not perceive the notes as existing [Russell] |
| Full Idea: When, after hearing the notes of a melody, I perceive the melody, the notes are not presented as still existing. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.31) | |
| A reaction: This is a good example, supporting Meinong's idea that we focus on 'intentional objects', rather than actual objects. |
| 21535 | Objects only exist if they 'occupy' space and time [Russell] |
| Full Idea: Only those objects exist which have to particular parts of space and time the special relation of 'occupying' them. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.29) | |
| A reaction: He excepts space and time themselves. Clearly this doesn't advance our understanding much, but it points to a priority in our normal conceptual scheme. Is Russell assuming absolute space and time? |
| 21533 | Contingency arises from tensed verbs changing the propositions to which they refer [Russell] |
| Full Idea: Contingency derives from the fact that a sentence containing a verb in the present tense - or sometimes in the past or the future - changes its meaning continually as the present changes, and stands for different propositions at different times. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.26) | |
| A reaction: This immediately strikes me as a bad example of the linguistic approach to philosophy. As if we (like any animal) didn't have an apprehension prior to any language that most parts of experience are capable of change. |
| 21537 | I assume we perceive the actual objects, and not their 'presentations' [Russell] |
| Full Idea: I prefer to advocate ...that the object of a presentation is the actual external object itself, and not any part of the presentation at all. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.33) | |
| A reaction: Although I am a fan of the robust realism usually favoured by Russell, I think he is wrong. I take Russell to be frightened that once you take perception to be of 'presentations' rather than things, there is a slippery slope to anti-realism. Not so. |
| 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. |
| 21532 | Full empiricism is not tenable, but empirical investigation is always essential [Russell] |
| Full Idea: Although empiricism as a philosophy does not appear to be tenable, there is an empirical manner of investigating, which should be applied in every subject-matter | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.22) | |
| A reaction: Given that early Russell loads his ontology with properties and propositions, this should come as no surprise, even if J.S. Mill was his godfather. |
| 21542 | Do incorrect judgements have non-existent, or mental, or external objects? [Russell] |
| Full Idea: Correct judgements have a transcendent object; but with regard to incorrect judgements, it remains to examine whether 1) the object is immanent, 2) there is no object, or 3) the object is transcendent. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.67) | |
| A reaction: Why is it that only Russell seems to have taken this problem seriously? Its solution gives the clearest possible indicator of how the mind relates to the world. |
| 21541 | The complexity of the content correlates with the complexity of the object [Russell] |
| Full Idea: Every property of the object seems to demand a strictly correlative property of the content, and the content, therefore, must have every complexity belonging to the object. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.55) | |
| A reaction: This claim gives a basis for his 'congruence' account of the correspondence theory of truth. It strikes me as false. If I talk of the 'red red robin', I don't mention the robin's feet. He ignores the psychological selection we make in abstraction. |
| 21543 | If p is false, then believing not-p is knowing a truth, so negative propositions must exist [Russell] |
| Full Idea: If p is a false affirmative proposition ...then it seems obvious that if we believe not-p we do know something true, so belief in not-p must be something which is not mere disbelief. This proves that there are negative propositions. | |
| From: Bertrand Russell (Meinong on Complexes and Assumptions [1904], p.75) | |
| A reaction: This evidently assumes excluded middle, but is none the worse for that. But it sounds suspiciously like believing there is no rhinoceros in the room. Does such a belief require a fact? |