1994 | What Required for Foundation for Maths? |
p.405-1 | p.405 | 17774 | Definitions make our intuitions mathematically useful |
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) |
p.405-1 | p.405 | 17775 | If proof and definition are central, then mathematics needs and possesses foundations |
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) |
p.405-1 | p.405 | 17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly |
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. |
p.405-2 | p.405 | 17777 | Foundations need concepts, definition rules, premises, and proof rules |
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) |
p.405-2 | p.405 | 17773 | Proof shows that it is true, but also why it must be true |
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. |
p.406-2 | p.406 | 17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures |
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) |
p.406-2 | p.406 | 17778 | Axiomatiation relies on isomorphic structures being essentially the same |
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. |
p.407-1 | p.407 | 17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects |
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? |
p.407-2 | p.407 | 17782 | Greek quantities were concrete, and ratio and proportion were their science |
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] |
p.407-2 | p.407 | 17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' |
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) |
p.408-2 | p.408 | 17785 | Real numbers as abstracted objects are now treated as complete ordered fields |
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). |
p.408-2 | p.408 | 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields |
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. |
p.410-1 | p.410 | 17786 | The mainstream of modern logic sees it as a branch of mathematics |
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'. |
p.410-2 | p.410 | 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation |
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. |
p.411-2 | p.411 | 17788 | First-order logic only has its main theorems because it is so weak |
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). |
p.412-1 | p.412 | 17791 | Only second-order logic can capture mathematical structure up to isomorphism |
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) |
p.412-1 | p.412 | 17790 | No Löwenheim-Skolem logic can axiomatise real analysis |
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) |
p.412-1 | p.412 | 17789 | No logic which can axiomatise arithmetic can be compact or complete |
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). |
p.412-1 | p.412 | 17792 | 1st-order PA is only interesting because of results which use 2nd-order PA |
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) |
p.412-1 | p.412 | 17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness |
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) |
p.412-2 | p.412 | 17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics |
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]. |
p.413-2 | p.413 | 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation |
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. |
p.413-2 | p.413 | 17796 | There is a semi-categorical axiomatisation of set-theory |
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. |
p.414-2 | p.414 | 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them |
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. |
p.414-2 | p.414 | 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size |
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) |
p.414-2 | p.414 | 17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers |
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) |
p.414-2 | p.414 | 17797 | Cantor extended the finite (rather than 'taming the infinite') |
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) |
p.415-1 | p.415 | 17802 | We don't translate mathematics into set theory, because it comes embodied in that way |
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. |
p.415-2 | p.415 | 17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms |
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) |
p.415-2 | p.415 | 17803 | Limitation of size is part of the very conception of a set |
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) |
p.416-1 | p.416 | 17805 | Set theory is not just another axiomatised part of mathematics |
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) |