47 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) |
| 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) |
| 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). |
| 17000 | We might fix identities for small particulars, but it is utopian to hope for such things [Kripke] |
| Full Idea: Maybe strict identity only applies to the particulars (the molecules) in a case of vague identity. …It seems, however, utopian to suppose that we will ever reach a level of ultimate, basic particulars for which identity relations are never vague. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 18) | |
| A reaction: I agree with this. Ladyman and Ross laugh at the unscientific picture found in dreams of 'simples'. |
| 11868 | A different piece of wood could have been used for that table; constitution isn't identity [Wiggins on Kripke] |
| Full Idea: Could the artificer not, when he made the table, have taken other pieces? Surely he could. [n37: I venture to think that Kripke's argument in note 56 for the necessity of constitution depends on treating constitution as if it were identity]. | |
| From: comment on Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 56) by David Wiggins - Sameness and Substance Renewed 4.11 | |
| A reaction: Suppose the craftsman completed the table, then changed a piece of wood in it for some reason. Has he now made a second table and destroyed the first one? Wiggins seems to be right. |
| 17044 | A relation can clearly be reflexive, and identity is the smallest reflexive relation [Kripke] |
| Full Idea: Some philosophers have thought that a relation, being essentially two-termed, cannot hold between a thing and itself. This position is plainly absurd ('he is his own worst enemy'). Identity is nothing but the smallest reflexive relation. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 50) | |
| A reaction: I have no idea what 'smallest' means here. I can't be 'to the left of myself', so not all of my relations can be reflexive. I just don't understand what it means to say something is 'identical with itself'. You've got the thing - what have you added? |
| 16999 | A vague identity may seem intransitive, and we might want to talk of 'counterparts' [Kripke] |
| Full Idea: When the identity relation is vague, it may seem intransitive; a claim of apparent identity may yield an apparent non-identity. Some sort of 'counterpart' notion may have some utility here. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 18) | |
| A reaction: He firmly rejects the full Lewis apparatus of counterparts. The idea would be that a river at different times had counterpart relations, not strict identity. I like the word 'same' for this situation. Most worldly 'identity' is intransitive. |
| 17058 | What many people consider merely physically necessary I consider completely necessary [Kripke] |
| Full Idea: My third lecture suggests that a good deal of what contemporary philosophy regards as mere physical necessity is actually necessary tout court. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], Add (g)) | |
| A reaction: He avoids the term 'metaphysically necessary', which most people would not use for this point. |
| 4970 | What is often held to be mere physical necessity is actually metaphysical necessity [Kripke] |
| Full Idea: My third lecture suggests that a good deal of what contemporary philosophy regards as mere physical necessity is actually necessary 'tout court'. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], Add (g)) | |
| A reaction: This huge claim rides in on the back of Kripke's very useful clarifications. It is the 'new essentialism', and seems to me untenable in this form. There is no answer to Hume's request for evidence of necessity. Why can't essences (and laws) change? |
| 17059 | Unicorns are vague, so no actual or possible creature could count as a unicorn [Kripke] |
| Full Idea: If the unicorn myth is supposed to be a particular species, with insufficient internal structure to determine it uniquely, then there is no actual or possible species of which we can say that it would have been the species of unicorns. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], Add (a)) | |
| A reaction: Dummett and Rumfitt discuss this proposal elsewhere. |
| 4950 | Possible worlds are useful in set theory, but can be very misleading elsewhere [Kripke] |
| Full Idea: The apparatus of possible worlds has (I hope) been very useful as far as the set-theoretic model-theory of quantified modal logic is concerned, but has encouraged philosophical pseudo-problems and misleading pictures. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 15) | |
| A reaction: This is presumably a swipe at David Lewis, who claims possible worlds are real. The fact that the originator of possible worlds sees them as unproblematic doesn't mean they are. Fine if they are a game, but if they assert truth, they need a metaphysics. |
| 17003 | Kaplan's 'Dthat' is a useful operator for transforming a description into a rigid designation [Kripke] |
| Full Idea: It is useful to have an operator which transforms each description into a term which rigidly designates the object actually satisfying the description. David Kaplan has proposed such an operator and calls it 'Dthat'. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 22) |
| 9221 | The best known objection to counterparts is Kripke's, that Humphrey doesn't care if his counterpart wins [Kripke, by Sider] |
| Full Idea: The most famous objection to counterparts is Kripke's objection that Hubert Humphrey wouldn't care if he thought that his counterpart might have won the 1972 election. He wishes that he had won it. | |
| From: report of Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 12) by Theodore Sider - Reductive Theories of Modality 3.10 | |
| A reaction: Like Sider, I find this unconvincing. If there is a world in which I don't exist, but my very close counterpart does (say exactly me, but with a finger missing), I am likely to care more about such a person than about complete strangers. |
| 17052 | The a priori analytic truths involving fixing of reference are contingent [Kripke] |
| Full Idea: If statements whose a priori truth is known via the fixing of a reference are counted as analytic, then some analytic truths are contingent. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 63) |
| 4969 | I regard the mind-body problem as wide open, and extremely confusing [Kripke] |
| Full Idea: I regard the mind-body problem as wide open, and extremely confusing. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 77) | |
| A reaction: Kripke opposes reductive physicalism, but is NOT committed to dualism. He seems to be drawn to Davidson or Nagel (see his note 73). I think his discussion of contingent mind-brain identity is confused. |
| 4956 | A description may fix a reference even when it is not true of its object [Kripke] |
| Full Idea: In some cases an object may be identified, and the reference of a name fixed, using a description which may turn out to be false of its object. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 34) | |
| A reaction: This is clearly possible. Someone could be identified as 'the criminal' when they were actually innocent. Nevertheless, how do you remember which person was baptised 'Aristotle' if you don't hang on to a description, even a false one? |
| 17032 | Even if Gödel didn't produce his theorems, he's still called 'Gödel' [Kripke] |
| Full Idea: If a Gödelian fraud were exposed, Gödel would no longer be called 'the author of the incompleteness theorem', but he would still be called 'Gödel'. The description, therefore, does not abbreviate the name. | |
| From: Saul A. Kripke (Naming and Necessity notes and addenda [1972], note 37) | |
| A reaction: Clearly we can't make the description a necessary fact about Gödel, but that doesn't invalidate the idea that successful reference needs some description. E.g. Gödel is a person. |
| 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. |