46 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) |
| 5331 | You can't infer that because you have a hidden birth-mark, everybody else does [Ayer] |
| Full Idea: My knowing that I had a hidden birth-mark would not entitle me to infer with any great degree of confidence that the same was true of everybody else. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.E) | |
| A reaction: This is the notorious 'induction from a single case' which was used by Mill to prove that other minds exist. It is a very nice illustration of the weakness of arguments from analogy. Probably analogy on its own is useless, but is a key part of induction. |
| 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) |
| 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). |
| 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) |
| 2611 | It is currently held that quantifying over something implies belief in its existence [Ayer] |
| Full Idea: It is currently held that we are committed to a belief in the existence of anything over which we quantify. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], IX.C) |
| 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). |
| 16520 | We see properties necessary for a kind (in the definition), but not for an individual [Ayer] |
| Full Idea: We can significantly ask what properties it is necessary for something to possess in order to be a thing of such and such a kind, since that asks what properties enter into the definition of the kind. But there is no such definition of the individual. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], 9.A.5) | |
| A reaction: [Quoted, not surprisingly, by Wiggins] Illuminating. If essence is just about necessary properties, I begin to see why the sortal might be favoured. I take it to concern explanatory mechanisms, and hence the individual. |
| 2613 | The theory of other minds has no rival [Ayer] |
| Full Idea: The theory that other people besides oneself have mental states is one that has no serious rival. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.E) | |
| A reaction: See 3463, where Searle says there is no such thing as our "theory" about other minds. In a science fiction situation (see 'Blade Runner'), this unrivalled theory could quickly unravel. It could even be a fact that you are the only humanoid with a mind. |
| 5328 | Originally I combined a mentalistic view of introspection with a behaviouristic view of other minds [Ayer] |
| Full Idea: In 1936 I combined a mentalistic analysis of the propositions in which one attributes experiences to oneself with a behaviouristic analysis of the propositions in which one attributes experiences to others. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.D) | |
| A reaction: He then criticises his view for inconsistency. Ryle preferred a behaviouristic account of introspection, but Ayer calls this 'ridiculous'. Ayer hunts for a compromise, but then settles for the right answer, which makes mentalism the 'best explanation'. |
| 5330 | Physicalism undercuts the other mind problem, by equating experience with 'public' brain events [Ayer] |
| Full Idea: The acceptance of physicalism undercuts the other minds problem by equating experiences with events in the brain, which are publicly observable. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.E) | |
| A reaction: It strikes me that if we could actually observe the operations of one another's brains, a great many of the problems of philosophy would never have appeared in the first place. Imagine a transparent skull and brain, with coloured waves moving through it. |
| 5326 | Qualia must be united by a subject, because they lead to concepts and judgements [Ayer] |
| Full Idea: The ground for thinking that qualia are only experiences because they relate to a unifying subject is that they have to be identified, by being brought under concepts, and giving rise to judgements which usually go beyond them. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.B) | |
| A reaction: Thus one of Hume's greatest fans gives the clearest objection to Hume. It strikes me as a very powerful objection, better than anything Carruthers offers (1394,1395,1396). The conceptual element is very hard to disentangle from the qualia. |
| 5325 | Is something an 'experience' because it relates to other experiences, or because it relates to a subject? [Ayer] |
| Full Idea: Is the character of being an item of experience one that can accrue to a quale through its relation to other qualia, or must it consist in a relation to a subject, which is conscious of these elements and distinct from them? | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.B) | |
| A reaction: When nicely put like this, it is hard to see how qualia could be experiences just because they relate to one another. It begs the question of what is causing the relationship. There seems to be a Cogito-like assumption of a thinker. |
| 5324 | Bodily identity and memory work together to establish personal identity [Ayer] |
| Full Idea: In general the two criteria of memory and bodily identity work together. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.B) | |
| A reaction: This seems better than any simplistic one-criterion approach. In life we use different criteria for our own identity, as when dreaming, or waking with a hangover, or wondering if we are dead after an accident. |
| 5322 | Self-consciousness is not basic, because experiences are not instrinsically marked with ownership [Ayer] |
| Full Idea: Self-consciousness is not a primitive datum, or in other words the observer's experiences are not intrinsically marked as his own. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.A) | |
| A reaction: This is a very Humean, ruthlessly empiricist view of the matter. Plenty of philosophers (existentialists, or Charles Taylor) would say that our experiences have our interests or values built into them. Why are they experiences, and not just events? |
| 5327 | Temporal gaps in the consciousness of a spirit could not be bridged by memories [Ayer] |
| Full Idea: If there were temporal gaps in the consciousness of disembodied spirits, the occurrences of memory-experiences would not be sufficient to bridge them. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.C) | |
| A reaction: Ayer is very sympathetic to the idea that the body is a key ingredient in personal identity. Without a body, there would be no criteria at all for the continuity of a spirit which lost consciousness for a while, since consciousness is all it is. |
| 5329 | Why shouldn't we say brain depends on mind? Better explanation! [Ayer] |
| Full Idea: If mind and brain exactly correspond we have as good ground for saying the brain depends on the mind as the other way round; if predominance is given to the brain, the reason is that it fits into a wider explanatory system. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], §VI.D) | |
| A reaction: A small but significant point. If an 'identity' theory is to be developed, then this step in the argument has to be justified. It is tempting here to move to the eliminativist view, because we no longer have to worry about a 'direction of priority'. |
| 2610 | Talk of propositions is just shorthand for talking about equivalent sentences [Ayer] |
| Full Idea: Our talk of propositions should not be regarded as anything more than a concise way of talking about equivalent sentences. | |
| From: A.J. Ayer (The Central Questions of Philosophy [1973], IX.C) | |
| A reaction: Wrong, though I can see why he says it. We struggle to express difficult propositions by offering several similar (but not equivalent) sentences. What is the criterion for deciding his 'equivalence'? |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |