49 ideas
| 17774 | Definitions make our intuitions mathematically useful [Mayberry] |
| Full Idea: Definition provides us with the means for converting our intuitions into mathematically usable concepts. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17773 | Proof shows that it is true, but also why it must be true [Mayberry] |
| Full Idea: When you have proved something you know not only that it is true, but why it must be true. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) | |
| A reaction: Note the word 'must'. Presumably both the grounding and the necessitation of the truth are revealed. |
| 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry] |
| Full Idea: Set theory cannot be an axiomatic theory, because the very notion of an axiomatic theory makes no sense without it. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: This will come as a surprise to Penelope Maddy, who battles with ways to accept the set theory axioms as the foundation of mathematics. Mayberry says that the basic set theory required is much more simple and intuitive. |
| 17796 | There is a semi-categorical axiomatisation of set-theory [Mayberry] |
| Full Idea: We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2) | |
| A reaction: I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory. |
| 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry] |
| Full Idea: The (misnamed!) Axiom of Infinity expresses Cantor's fundamental assumption that the species of natural numbers is finite in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry] |
| Full Idea: The idea of 'generating' sets is only a metaphor - the existence of the hierarchy is established without appealing to such dubious notions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) | |
| A reaction: Presumably there can be a 'dependence' or 'determination' relation which does not involve actual generation. |
| 17803 | Limitation of size is part of the very conception of a set [Mayberry] |
| Full Idea: Our very notion of a set is that of an extensional plurality limited in size. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17786 | The mainstream of modern logic sees it as a branch of mathematics [Mayberry] |
| Full Idea: In the mainstream tradition of modern logic, beginning with Boole, Peirce and Schröder, descending through Löwenheim and Skolem to reach maturity with Tarski and his school ...saw logic as a branch of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-1) | |
| A reaction: [The lesser tradition, of Frege and Russell, says mathematics is a branch of logic]. Mayberry says the Fregean tradition 'has almost died out'. |
| 17788 | First-order logic only has its main theorems because it is so weak [Mayberry] |
| Full Idea: First-order logic is very weak, but therein lies its strength. Its principle tools (Compactness, Completeness, Löwenheim-Skolem Theorems) can be established only because it is too weak to axiomatize either arithmetic or analysis. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.411-2) | |
| A reaction: He adds the proviso that this is 'unless we are dealing with structures on whose size we have placed an explicit, finite bound' (p.412-1). |
| 17791 | Only second-order logic can capture mathematical structure up to isomorphism [Mayberry] |
| Full Idea: Second-order logic is a powerful tool of definition: by means of it alone we can capture mathematical structure up to isomorphism using simple axiom systems. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 8195 | Undecidable statements result from quantifying over infinites, subjunctive conditionals, and the past tense [Dummett] |
| Full Idea: I once wrote that there are three linguistic devices that make it possible for us to frame undecidable statements: quantification over infinity totalities, as expressed by word such as 'never'; the subjunctive conditional form; and the past tense. | |
| From: Michael Dummett (Truth and the Past [2001], 4) | |
| A reaction: Dummett now repudiates the third one. Statements containing vague concepts also appear to be undecidable. Personally I have no problems with deciding (to a fair extent) about 'never x', and 'if x were true', and 'it was x'. |
| 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry] |
| Full Idea: The 'logica magna' [of the Fregean tradition] has quantifiers ranging over a fixed domain, namely everything there is. In the Boolean tradition the domains differ from interpretation to interpretation. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.410-2) | |
| A reaction: Modal logic displays both approaches, with different systems for global and local domains. |
| 17790 | No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry] |
| Full Idea: No logic which can axiomatize real analysis can have the Löwenheim-Skolem property. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry] |
| Full Idea: The purpose of a 'classificatory' axiomatic theory is to single out an otherwise disparate species of structures by fixing certain features of morphology. ...The aim is to single out common features. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) |
| 17778 | Axiomatiation relies on isomorphic structures being essentially the same [Mayberry] |
| Full Idea: The central dogma of the axiomatic method is this: isomorphic structures are mathematically indistinguishable in their essential properties. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.406-2) | |
| A reaction: Hence it is not that we have to settle for the success of a system 'up to isomorphism', since that was the original aim. The structures must differ in their non-essential properties, or they would be the same system. |
| 17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry] |
| Full Idea: The purpose of what I am calling 'eliminatory' axiomatic theories is precisely to eliminate from mathematics those peculiar ideal and abstract objects that, on the traditional view, constitute its subject matter. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-1) | |
| A reaction: A very interesting idea. I have a natural antipathy to 'abstract objects', because they really mess up what could otherwise be a very tidy ontology. What he describes might be better called 'ignoring' axioms. The objects may 'exist', but who cares? |
| 17789 | No logic which can axiomatise arithmetic can be compact or complete [Mayberry] |
| Full Idea: No logic which can axiomatise arithmetic can be compact or complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) | |
| A reaction: I take this to be because there are new truths in the transfinite level (as well as the problem of incompleteness). |
| 8194 | Surely there is no exact single grain that brings a heap into existence [Dummett] |
| Full Idea: There is surely no number n such that "n grains of sand do not make a heap, although n+1 grains of sand do" is true. | |
| From: Michael Dummett (Truth and the Past [2001], 4) | |
| A reaction: It might be argued that there is such a number, but no human being is capable of determing it. Might God know the value of n? On the whole Dummett's view seems the most plausible. |
| 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry] |
| Full Idea: We eliminate the real numbers by giving an axiomatic definition of the species of complete ordered fields. These axioms are categorical (mutually isomorphic), and thus are mathematically indistinguishable. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: Hence my clever mathematical friend says that it is a terrible misunderstanding to think that mathematics is about numbers. Mayberry says the reals are one ordered field, but mathematics now studies all ordered fields together. |
| 17782 | Greek quantities were concrete, and ratio and proportion were their science [Mayberry] |
| Full Idea: Quantities for Greeks were concrete things - lines, surfaces, solids, times, weights. At the centre of their science of quantity was the beautiful theory of ratio and proportion (...in which the notion of number does not appear!). | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) | |
| A reaction: [He credits Eudoxus, and cites Book V of Euclid] |
| 17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry] |
| Full Idea: The abstract objects of modern mathematics, the real numbers, were invented by the mathematicians of the seventeenth century in order to simplify and to generalize the Greek science of quantity. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.407-2) |
| 17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry] |
| Full Idea: In Cantor's new vision, the infinite, the genuine infinite, does not disappear, but presents itself in the guise of the absolute, as manifested in the species of all sets or the species of all ordinal numbers. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17797 | Cantor extended the finite (rather than 'taming the infinite') [Mayberry] |
| Full Idea: We may describe Cantor's achievement by saying, not that he tamed the infinite, but that he extended the finite. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2) |
| 17775 | If proof and definition are central, then mathematics needs and possesses foundations [Mayberry] |
| Full Idea: If we grant, as surely we must, the central importance of proof and definition, then we must also grant that mathematics not only needs, but in fact has, foundations. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) |
| 17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry] |
| Full Idea: The ultimate principles upon which mathematics rests are those to which mathematicians appeal without proof; and the primitive concepts of mathematics ...themselves are grasped directly, if grasped at all, without the mediation of definition. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-1) | |
| A reaction: This begs the question of whether the 'grasping' is purely a priori, or whether it derives from experience. I defend the latter, and Jenkins puts the case well. |
| 17777 | Foundations need concepts, definition rules, premises, and proof rules [Mayberry] |
| Full Idea: An account of the foundations of mathematics must specify four things: the primitive concepts for use in definitions, the rules governing definitions, the ultimate premises of proofs, and rules allowing advance from premises to conclusions. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.405-2) |
| 17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry] |
| Full Idea: No axiomatic theory, formal or informal, of first or of higher order can logically play a foundational role in mathematics. ...It is obvious that you cannot use the axiomatic method to explain what the axiomatic method is. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2) |
| 17792 | 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry] |
| Full Idea: The sole theoretical interest of first-order Peano arithmetic derives from the fact that it is a first-order reduct of a categorical second-order theory. Its axioms can be proved incomplete only because the second-order theory is categorical. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry] |
| Full Idea: If we did not know that the second-order axioms characterise the natural numbers up to isomorphism, we should have no reason to suppose, a priori, that first-order Peano Arithmetic should be complete. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-1) |
| 17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry] |
| Full Idea: The idea that set theory must simply be identified with first-order Zermelo-Fraenkel is surprisingly widespread. ...The first-order axiomatic theory of sets is clearly inadequate as a foundation of mathematics. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.412-2) | |
| A reaction: [He is agreeing with a quotation from Skolem]. |
| 17802 | We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry] |
| Full Idea: One does not have to translate 'ordinary' mathematics into the Zermelo-Fraenkel system: ordinary mathematics comes embodied in that system. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-1) | |
| A reaction: Mayberry seems to be a particular fan of set theory as spelling out the underlying facts of mathematics, though it has to be second-order. |
| 17805 | Set theory is not just another axiomatised part of mathematics [Mayberry] |
| Full Idea: The fons et origo of all confusion is the view that set theory is just another axiomatic theory and the universe of sets just another mathematical structure. ...The universe of sets ...is the world that all mathematical structures inhabit. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.416-1) |
| 8190 | Intuitionists rely on the proof of mathematical statements, not their truth [Dummett] |
| Full Idea: The intuitionist account of the meaning of mathematical statements does not employ the notion of a statement's being true, but only that of something's being a proof of the statement. | |
| From: Michael Dummett (Truth and the Past [2001], 2) | |
| A reaction: I remain unconvinced that anyone could give an account of proof that didn't discreetly employ the notion of truth. What are we to make of "we suspect this is true, but no one knows how to prove it?" (e.g. Goldbach's Conjecture). |
| 8198 | A 'Cambridge Change' is like saying 'the landscape changes as you travel east' [Dummett] |
| Full Idea: The idea of 'Cambridge Change' is like saying 'the landscape changes as you travel east'. | |
| From: Michael Dummett (Truth and the Past [2001], 5) | |
| A reaction: The phrase was coined in Oxford. It is a useful label with which realists can insult solipsists, idealists and other riff-raff. Four Dimensionalists seem to see time in this way. Events sit there, and we travel past them. But there are indexical events. |
| 8192 | I no longer think what a statement about the past says is just what can justify it [Dummett] |
| Full Idea: In distinguishing between what can establish a statement about the past as true and what it is that that statement says, we are repudiating antirealism about the past. | |
| From: Michael Dummett (Truth and the Past [2001], 3) | |
| A reaction: This is a late shift of ground from the champion of antirealism. If Dummett's whole position is based on a 'justificationist' theory of meaning, he must surely have a different theory of meaning now for statements about the past? |
| 17785 | Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry] |
| Full Idea: The abstractness of the old fashioned real numbers has been replaced by generality in the modern theory of complete ordered fields. | |
| From: John Mayberry (What Required for Foundation for Maths? [1994], p.408-2) | |
| A reaction: In philosophy, I'm increasingly thinking that we should talk much more of 'generality', and a great deal less about 'universals'. (By which I don't mean that redness is just the set of red things). |
| 8199 | The existence of a universe without sentience or intelligence is an unintelligible fantasy [Dummett] |
| Full Idea: The existence of a universe from which sentience was permanently absent is an unintelligible fantasy. What exists is what can be known to exist. What is true is what can be known to be true. Reality is what can be experienced and known. | |
| From: Michael Dummett (Truth and the Past [2001], 5) | |
| A reaction: This strikes me as nonsense. The fact that we cannot think about a universe without introducing a viewpoint does not mean that we cannot 'intellectually imagine' its existence devoid of viewpoints. Nothing could ever experience a star's interior. |
| 21833 | Research suggest that we overrate conscious experience [Flanagan] |
| Full Idea: The emerging consensus is that we probably overrate the power of conscious experience in our lives. Freud, of course, said the same thing for different reasons. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Ontology') | |
| A reaction: [He cites Pockett, Banks and Gallagher 2006]. Freud was concerned with big deep secrets, but the modern view concerns ordinary decisions and perceptions. An important idea, which should incline us all to become Nietzscheans. |
| 21834 | Sensations may be identical to brain events, but complex mental events don't seem to be [Flanagan] |
| Full Idea: There is still some hope for something like identity theory for sensations. But almost no one believes that strict identity theory will work for more complex mental states. Strict identity is stronger than type neurophysicalism. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Ontology') | |
| A reaction: It is so hard to express the problem. What needs to be explained? How can one bunch of neurons represent many different things? It's not like computing. That just transfers the data to brains, where the puzzling stuff happens. |
| 8193 | Verification is not an individual but a collective activity [Dummett] |
| Full Idea: Verification is not an individual but a collective activity. | |
| From: Michael Dummett (Truth and the Past [2001], 3) | |
| A reaction: This generates problems. Are deceased members of the community included? (Yes, says Dummett). If someone speaks to angels (Blake!), do they get included? Is a majority necessary? What of weird loners? Etc. |
| 8189 | Truth-condition theorists must argue use can only be described by appeal to conditions of truth [Dummett] |
| Full Idea: To demonstrate the necessity of a truth-conditional theory of meaning, a proponent of such a theory must argue that use cannot be described without appeal to the conditions for the truth of statements. | |
| From: Michael Dummett (Truth and the Past [2001], 1) | |
| A reaction: Unlike Dummett, I find that argument rather appealing. How do you decide the possible or appropriate use for a piece of language, if you don't already know what it means. Basing it all on social conventions means it could be meaningless ritual. |
| 8191 | The truth-conditions theory must get agreement on a conception of truth [Dummett] |
| Full Idea: It is not enough for the truth-condition theorist to argue that we need the concept of truth: he must show that we should have the same conception of truth that he has. | |
| From: Michael Dummett (Truth and the Past [2001], 2) | |
| A reaction: Davidson invites us to accept Tarski's account of truth. It invites the question of what the theory would be like with a very robust correspondence account of truth, or a flabby rather subjective coherence view, or the worst sort of pragmatic view. |
| 21837 | Morality is normative because it identifies best practices among the normal practices [Flanagan] |
| Full Idea: Morality is 'normative' in the sense that it consists of the extraction of ''good' or 'excellent' practices from common practices. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 4 'Naturalism') |
| 21830 | For Darwinians, altruism is either contracts or genetics [Flanagan] |
| Full Idea: Two explanations came forward in the neo-Darwinian synthesis. Altruism is either 1) person-based reciprocal altruism, or 2) gene-based kin altruism. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 2 'Darwin') | |
| A reaction: Flanagan obviously thinks there is also 'genuine psychological atruism'. Presumably we don't explain mathematics or music or the desire to travel as either contracts or genetics, so we have other explanations available. |
| 21835 | We need Eudaimonics - the empirical study of how we should flourish [Flanagan] |
| Full Idea: It would be nice if I could advance the case for Eudaimonics - empirical enquiry into the nature, causes, and constituents of flourishing, …and the case for some ways of living and being as better than others. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 4 'Normative') | |
| A reaction: Things seem to be moving in that direction. Lots of statistics about happiness have been appearing. |
| 21831 | Alienation is not finding what one wants, or being unable to achieve it [Flanagan] |
| Full Idea: What Marx called 'alienation' is the widespread condition of not being able to discover what one wants, or not being remotely positioned to achieve. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 2 'Expanding') | |
| A reaction: I took alienation to concern people's relationship to the means of production in their trade. On Flanagan's definition I would expect almost everyone aged under 20 to count as alienated. |
| 8197 | Maybe past (which affects us) and future (which we can affect) are both real [Dummett] |
| Full Idea: Maybe both the past and the future are real, determined by our current temporal perspective. Past is then events capable of having a causal influence upon events near us, and future is events we can affect, but from which we receive no information. | |
| From: Michael Dummett (Truth and the Past [2001], 5) | |
| A reaction: This is the Four-Dimensional view, which is opposed to Presentism. Might immediate unease is that it gives encouragement to fortune-tellers, whom I have always dismissed with 'You can't see the future, because it doesn't exist'. |
| 8196 | The present cannot exist alone as a mere boundary; past and future truths are rendered meaningless [Dummett] |
| Full Idea: The idea that only the present is real cannot be sustained. St Augustine pointed out that the present has no duration; it is a mere boundary between past and future, and dependent on them. It also denies truth-value to statements about past or future. | |
| From: Michael Dummett (Truth and the Past [2001], 5) | |
| A reaction: To defend Presentism, I suspect that one must focus entirely on the activities of consciousness and short-term memory. All truths, of past or future, must refer totally to such mental events. But what could an event be if there is no enduring time? |
| 21832 | Buddhists reject God and the self, and accept suffering as key, and liberation through wisdom [Flanagan] |
| Full Idea: Buddhism rejected the idea of a creator God, and the unchanging self [atman]. They accept the appearance-reality distinction, reward for virtue [karma], suffering defining our predicament, and that liberation [nirvana] is possible through wisdom. | |
| From: Owen Flanagan (The Really Hard Problem [2007], 3 'Buddhism') | |
| A reaction: [Compressed] Flanagan is an analytic philosopher and a practising Buddhist. Looking at a happiness map today which shows Europeans largely happy, and Africans largely miserable, I can see why they thought suffering was basic. |