48 ideas
2557 | Analytical philosophy seems to have little interest in how to tell a good analysis from a bad one [Rorty] |
Full Idea: There is nowadays little attempt to bring "analytic philosophy" to self-consciousness by explaining how to tell a successful from an unsuccessful analysis. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.1) |
2556 | Rational certainty may be victory in argument rather than knowledge of facts [Rorty] |
Full Idea: We can think of "rational certainty" as a matter of victory in argument rather than relation to an object known. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 3.4) |
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. |
4726 | Rorty seems to view truth as simply being able to hold one's view against all comers [Rorty, by O'Grady] |
Full Idea: Rorty seems to view truth as simply being able to hold one's view against all comers. | |
From: report of Richard Rorty (Philosophy and the Mirror of Nature [1980]) by Paul O'Grady - Relativism Ch.4 | |
A reaction: This may be a caricature of Rorty, but he certainly seems to be in the business of denying truth as much as possible. This strikes me as the essence of pragmatism, and as a kind of philosophical nihilism. |
2549 | For James truth is "what it is better for us to believe" rather than a correct picture of reality [Rorty] |
Full Idea: Truth is, in James' phrase, "what it is better for us to believe", rather than "the accurate representation of reality". | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) |
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) |
15312 | We get the idea of power by abstracting from ropes, magnets and electric shocks [Priestley] |
Full Idea: A rope sustains weight, a magnet attracts iron, a charged electrical jar gives a shock, and from these and other similar observations, we get the idea of power, universally and abstractly considered. | |
From: Joseph Priestley (Theological and other works [1790], p.191), quoted by Harré,R./Madden,E.H. - Causal Powers 9.II.B | |
A reaction: I agree with this, in that we appear to be observing powers directly, and are not observing something which can then be reduced to non-powers. Nature just can't be a set of inert structures, with forces 'imposed' on them. |
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). |
2548 | If knowledge is merely justified belief, justification is social [Rorty] |
Full Idea: If we have a Deweyan conception of knowledge, as what we are justified in believing, we will see "justification" as a social phenomenon. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Intro) | |
A reaction: I find this observation highly illuminating (though I probably need to study Dewey to understand it). There just is no absolute about whether someone is justified. How justified do you want to be? |
6599 | Knowing has no definable essence, but is a social right, found in the context of conversations [Rorty] |
Full Idea: If we see knowing not as having an essence, described by scientists or philosophers, but rather as a right, by current standards, to believe, then we see conversation as the ultimate context within which knowledge is to be understood. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], Ch.5), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
A reaction: This teeters towards ridiculous relativism (e.g. what if the conversation is among a group of fools? - Ah, there are no fools! Politically incorrect!). However, knowledge can be social, provided we are healthily elitist. Scientists know more than us. |
2566 | You can't debate about whether to have higher standards for the application of words [Rorty] |
Full Idea: The decision about whether to have higher than usual standards for the application of words like "true" or "good" or "red" is, as far as I can see, not a debatable issue. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.6) |
2553 | The mind is a property, or it is baffling [Rorty] |
Full Idea: All that is needed for the mind-body problem to be unintelligible is for us to be nominalist, to refuse firmly to hypostasize individual properties. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
A reaction: Edelman says the mind is a process rather than a property. It might vanish if the clockspeed was turned right down? Nominalism here sounds like behaviourism or instrumentalism. Would Dennett plead guilty? |
2550 | Pain lacks intentionality; beliefs lack qualia [Rorty] |
Full Idea: We can't define the mental as intentional because pains aren't about anything, and we can't define it as phenomenal because beliefs don't feel like anything. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.2) | |
A reaction: Nice, but simplistic? There is usually an intentional object for a pain, and the concepts which we use to build beliefs contain the residue of remembered qualia. It seems unlikely that any mind could have one without the other (even a computer). |
2554 | Is intentionality a special sort of function? [Rorty] |
Full Idea: Following Wittgenstein, we shall treat the intentional as merely a subspecies of the functional. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 1.3) | |
A reaction: Intriguing but obscure. Sounds wrong to me. The intentional refers to the content of thoughts, but function concerns their role. They have roles because they have content, so they can't be the same. |
2565 | Nature has no preferred way of being represented [Rorty] |
Full Idea: Nature has no preferred way of being represented. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.5) | |
A reaction: Tree rings accidentally represent the passing of the years. If God went back and started again would she or he opt for a 'preferred way'? |
2560 | Can meanings remain the same when beliefs change? [Rorty] |
Full Idea: For cooler heads there must be some middle view between "meanings remain and beliefs change" and "meanings change whenever beliefs do". | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.2) | |
A reaction: The second one seems blatanty false. How could we otherwise explain a change in belief? But obviously some changes in belief (e.g. about electrons) produce a change in meaning. |
2562 | A theory of reference seems needed to pick out objects without ghostly inner states [Rorty] |
Full Idea: The need to pick out objects without the help of definitions, essences, and meanings of terms produced, philosophers thought, a need for a "theory of reference". | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.3) | |
A reaction: Frege's was very perceptive in noting that meaning and reference are not the same. Whether we need a 'theory' of reference is unclear. It is worth describing how it occurs. |
2559 | Davidson's theory of meaning focuses not on terms, but on relations between sentences [Rorty] |
Full Idea: A theory of meaning, for Davidson, is not an assemblage of "analyses" of the meanings of individual terms, but rather an understanding of the inferential relations between sentences. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 6.1) | |
A reaction: Put that way, the influence of Frege on Davidson is obvious. Purely algebraic expressions can have inferential relations, using variables and formal 'sentences'. |
2558 | Since Hegel we have tended to see a human as merely animal if it is outside a society [Rorty] |
Full Idea: Only since Hegel have philosophers begun toying with the idea that the individual apart from his society is just one more animal. | |
From: Richard Rorty (Philosophy and the Mirror of Nature [1980], 4.3) |
15311 | Attraction or repulsion are not imparted to matter, but actually constitute it [Priestley] |
Full Idea: Attraction or repulsion appear to me not to be properly what is imparted to matter, but what really makes it what it is, in so much that, without it, it would be nothing at all. | |
From: Joseph Priestley (Theological and other works [1790], p.237), quoted by Harré,R./Madden,E.H. - Causal Powers 9.II.B | |
A reaction: This is music to the ears of anyone who thinks that powers are the fundamentals of nature (like me). |