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All the ideas for '67: Platonic Questions', 'Abstract Objects' and 'What Required for Foundation for Maths?'

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40 ideas

2. Reason / D. Definition / 2. Aims of Definition
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)
2. Reason / E. Argument / 6. Conclusive Proof
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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.
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.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
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)
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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)
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
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'.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
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).
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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)
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
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.
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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)
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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.
'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)
'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?
5. Theory of Logic / K. Features of Logics / 6. Compactness
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).
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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.
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
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)
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]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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)
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)
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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)
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.
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)
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)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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)
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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].
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.
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)
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
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).
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
How we refer to abstractions is much less clear than how we refer to other things [Rosen]
     Full Idea: It is unclear how we manage to refer determinately to abstract entities in a sense in which it is not unclear how we manage to refer determinately to other things.
     From: Gideon Rosen (Abstract Objects [2001], 'Way of Ex')
     A reaction: This is where problems of abstraction overlap with problems about reference in language. Can we have a 'baptism' account of each abstraction (even very large numbers)? Will descriptions do it? Do abstractions collapse into particulars when we refer?
15. Nature of Minds / A. Nature of Mind / 2. Psuche
When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch]
     Full Idea: The soul, when it has partaken of intelligence and reason and concord, is not merely a work but also a part of god and has come to be not by his agency but both from him as source and out of his substance.
     From: Plutarch (67: Platonic Questions [c.85], II.1001)
     A reaction: A most intriguing shift of view from earlier concepts of the psuché. How did this come about? This man is a pagan. The history is in the evolution of Platonism. See 'The Middle Platonists' by John Dillon. Davidson is also very impressed by reason.
18. Thought / E. Abstraction / 2. Abstracta by Selection
The Way of Abstraction used to say an abstraction is an idea that was formed by abstracting [Rosen]
     Full Idea: The simplest version of the Way of Abstraction would be to say that an object is abstract if it is a referent of an idea that was formed by abstraction, but this is wedded to an outmoded philosophy of mind.
     From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs')
     A reaction: This presumably refers to Locke, who wields the highly ambiguous term 'idea'. But if we sort out that ambiguity (by using modern talk of mental events, concepts and content?) we might reclaim the view. But do we have a 'genetic fallacy' here?
18. Thought / E. Abstraction / 5. Abstracta by Negation
Nowadays abstractions are defined as non-spatial, causally inert things [Rosen]
     Full Idea: If any characterization of the abstract deserves to be regarded as the modern standard one, it is this: an abstract entity is a non-spatial (or non-spatiotemporal) causally inert thing. This view presents a number of perplexities...
     From: Gideon Rosen (Abstract Objects [2001], 'Non-spat')
     A reaction: As indicated in other ideas, the problem is that some abstractions do seem to be located somewhere in space-time, and to have come into existence, and to pass away. I like 'to exist is to have causal powers'. See Ideas 5992 and 8300.
Chess may be abstract, but it has existed in specific space and time [Rosen]
     Full Idea: The natural view of chess is not that it is a non-spatiotemporal mathematical object, but that it was invented at a certain time and place, that it has changed over the years, and so on.
     From: Gideon Rosen (Abstract Objects [2001], 'Non-spat')
     A reaction: This strikes me as being undeniable, and being an incredibly important point. Logicians seem to want to subsume things like games into the highly abstract world of logic and numbers. In fact the direction of explanation should be reversed.
Sets are said to be abstract and non-spatial, but a set of books can be on a shelf [Rosen]
     Full Idea: It is thought that sets are abstract, abstract objects do not exist in space, so sets must not exist in space. But it is not unnatural to say that a set of books is located on a certain shelf in the library.
     From: Gideon Rosen (Abstract Objects [2001], 'Non-spat')
     A reaction: The arguments against non-spatiality of abstractions seem to me to be conclusive. Not being able to assign a location to the cosine function is on a par with not knowing where my thoughts are located in my brain.
18. Thought / E. Abstraction / 6. Abstracta by Conflation
Conflating abstractions with either sets or universals is a big claim, needing a big defence [Rosen]
     Full Idea: The Way of Conflation account of abstractions (identifying them sets or with universals) is now relatively rare. The claim sets or universals are the only abstract objects would amount to a substantive metaphysical thesis, in need of defence.
     From: Gideon Rosen (Abstract Objects [2001], 'Way of Con')
     A reaction: If you produce a concept like 'mammal' by psychological abstraction, you do seem to end up with a set of things with shared properties, so this approach is not silly. I can't think of any examples of abstractions which are not sets or universals.
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Functional terms can pick out abstractions by asserting an equivalence relation [Rosen]
     Full Idea: On Frege's suggestion, functional terms that pick out abstract expressions (such as 'direction' or 'equinumeral') have a typical form of f(a) = f(b) iff aRb, where R is an equivalence relation, a relation which is reflexive, symmetric and transitive.
     From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs')
     A reaction: [Wright and Hale are credited with the details] This has become the modern orthodoxy among the logically-minded. Examples of R are 'parallel' or 'just as many as'. It picks out an 'aspect', which isn't far from the old view.
Abstraction by equivalence relationships might prove that a train is an abstract entity [Rosen]
     Full Idea: It seems possible to define a train in terms of its carriages and the connection relationship, which would meet the equivalence account of abstraction, but demonstrate that trains are actually abstract.
     From: Gideon Rosen (Abstract Objects [2001], 'Way of Abs')
     A reaction: [Compressed. See article for more detail] A tricky example, but a suggestive line of criticism. If you find two physical objects which relate to one another reflexively, symmetrically and transitively, they may turn out to be abstract.