64 ideas
3099 | Inference is never a conscious process [Harman] |
Full Idea: Inference is never a conscious process. | |
From: Gilbert Harman (Thought [1973], 11.2) |
3077 | Reasoning might be defined in terms of its functional role, which is to produce knowledge [Harman] |
Full Idea: Reasoning could be treated as a functionally defined process that is partly defined in terms of its role in giving a person knowledge. | |
From: Gilbert Harman (Thought [1973], 3.6) |
3092 | If you believe that some of your beliefs are false, then at least one of your beliefs IS false [Harman] |
Full Idea: If a rational man believes he has at least some other false beliefs, it follows that a rational man knows that at least one of his beliefs is false (the one believed false, or this new belief). | |
From: Gilbert Harman (Thought [1973], 7.2) |
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) |
3093 | Any two states are logically linked, by being entailed by their conjunction [Harman] |
Full Idea: Any two states of affairs are logically connected, simply because both are entailed by their conjunction. | |
From: Gilbert Harman (Thought [1973], 8.1) |
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). |
3098 | Deductive logic is the only logic there is [Harman] |
Full Idea: Deductive logic is the only logic there is. | |
From: Gilbert Harman (Thought [1973], 10.4) |
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) |
3094 | You don't have to accept the conclusion of a valid argument [Harman] |
Full Idea: We may say "From P and If-P-then-Q, infer Q" (modus ponens), but there is no rule of acceptance to say that we should accept Q. Maybe we should stop believing P or If-P-then-Q rather than believe Q. | |
From: Gilbert Harman (Thought [1973], 10.1) |
3084 | Our underlying predicates represent words in the language, not universal concepts [Harman] |
Full Idea: The underlying truth-conditional structures of thoughts are language-dependent in the sense that underlying predicates represent words in the language rather than universal concepts common to all languages. | |
From: Gilbert Harman (Thought [1973], 6.3) |
3080 | Logical form is the part of a sentence structure which involves logical elements [Harman] |
Full Idea: The logical form of a sentence is that part of its structure that involves logical elements. | |
From: Gilbert Harman (Thought [1973], 5.2) |
3081 | A theory of truth in a language must involve a theory of logical form [Harman] |
Full Idea: Some sort of theory of logical form is involved in any theory of truth for a natural language. | |
From: Gilbert Harman (Thought [1973], 5.2) |
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) |
10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
Full Idea: Mathematics seems necessary because the real contents of mathematical statements are logical truths, which are necessary, and it seems a priori because logical truths really are a priori. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 10) | |
A reaction: Yablo says his logicism has a Kantian strain, because numbers and sets 'inscribed on our spectacles', but he takes a different view (in the present Idea) from Kant about where the necessity resides. Personally I am tempted by an a posteriori necessity. |
10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
Full Idea: Saying 'the number of Fs is 5', instead of using five quantifiers, puts the numeral in quantifiable position, which brings expressive advantages. 'There are more sheep in the field than cows' is an infinite disjunction, expressible in finite compass. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 08) | |
A reaction: See Hofweber with similar thoughts. This idea I take to be a key one in explaining many metaphysical confusions. The human mind just has a strong tendency to objectify properties, relations, qualities, categories etc. - for expression and for reasoning. |
10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
Full Idea: Objects like me have a few essential properties, and numerous accidental ones. Abstract objects are a different story. The intrinsic properties of the empty set are mostly essential. The relations of numbers are also mostly essential. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 01) |
10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
Full Idea: Our knowledge of concreta is a posteriori, but our knowledge of numbers, at least, has often been considered a priori. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 02) |
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). |
3100 | You have to reaffirm all your beliefs when you make a logical inference [Harman] |
Full Idea: Since inference is inference to the best total account, all your prior beliefs are relevant and your conclusion is everything you believe at the end. So, you constantly reaffirm your beliefs in inference. | |
From: Gilbert Harman (Thought [1973], 12.1) |
3089 | Only lack of imagination makes us think that 'cats are animals' is analytic [Harman] |
Full Idea: That 'cats are animals' is often cited as an analytic truth. But (as Putnam points out) the inability to imagine this false is just a lack of imagination. They might turn out to be radio-controlled plastic spies from Mars. | |
From: Gilbert Harman (Thought [1973], 6.7) |
3088 | Analyticity is postulated because we can't imagine some things being true, but we may just lack imagination [Harman] |
Full Idea: Analyticity is postulated to explain why we cannot imagine certain things being true. A better postulate is that we are not good at imagining things. | |
From: Gilbert Harman (Thought [1973], 6.7) |
3101 | Memories are not just preserved, they are constantly reinferred [Harman] |
Full Idea: I favour the inferential view of memory over the preservation view. …One constantly reinfers old beliefs. | |
From: Gilbert Harman (Thought [1973], 12.1) | |
A reaction: This has a grain of truth, but seems a distortion. An image of the old home floats into my mind when I am thinking about something utterly unconnected. When we search memory we may be inferring and explaining, but the same applies to searching images. |
3074 | People's reasons for belief are rarely conscious [Harman] |
Full Idea: The reasons for which people believe things are rarely conscious. | |
From: Gilbert Harman (Thought [1973], 2.2) | |
A reaction: Probably correct. The interesting bit is when they bring the beliefs into consciousness and scrutinise them rationally. Philosophers routinely overthrow their natural beliefs in this way. |
3097 | We don't distinguish between accepting, and accepting as evidence [Harman] |
Full Idea: There is no distinction between what we accept as evidence and whatever else we accept. | |
From: Gilbert Harman (Thought [1973], 10.4) |
6369 | In negative coherence theories, beliefs are prima facie justified, and don't need initial reasons [Harman, by Pollock/Cruz] |
Full Idea: According to Harman's negative coherence theory it is always permissible to adopt a new belief - any new belief; because beliefs are prima facie justified you do not need a reason for adopting a new belief. | |
From: report of Gilbert Harman (Thought [1973]) by J Pollock / J Cruz - Contemporary theories of Knowledge (2nd) §3.4.1 | |
A reaction: This must be placed alongside the fact that we don't usually choose our beliefs, but simply find ourselves believing because of the causal impact of evidence. This gives an unstated rational justification for any belief - something caused it. |
3096 | Coherence avoids scepticism, because it doesn't rely on unprovable foundations [Harman] |
Full Idea: Scepticism is undermined once it is seen that the relevant kind of justification is not a matter of derivation from basic principles but is rather a matter of showing that a view fits in well with other things we believe. | |
From: Gilbert Harman (Thought [1973], 10.4) | |
A reaction: I would (now) call myself a 'coherentist' about justification, and I agree with this. Coherent justification could not possibly deliver certainty, so it must be combined with fallibilism. |
3095 | Induction is an attempt to increase the coherence of our explanations [Harman] |
Full Idea: Induction is an attempt to increase the explanatory coherence of our view, making it more complete, less ad hoc, more plausible. | |
From: Gilbert Harman (Thought [1973], 10.2) |
3073 | We see ourselves in the world as a map [Harman] |
Full Idea: Our conception of ourselves in the world is more like a map than a story. | |
From: Gilbert Harman (Thought [1973], Pref) | |
A reaction: Dennett offer the 'story' view of the self (Ideas 7381 and 7382). How do we arbitrate this one? A story IS a sort of map. Maps can extend over time as well over space. I think the self is real, and is a location on a map, and the hero of a story. |
3076 | Defining dispositions is circular [Harman] |
Full Idea: There is no noncircular way to specify dispositions; for they are dispositions to behave given certain situations, and the situations must be include beliefs about the situation, and desires concerning it. | |
From: Gilbert Harman (Thought [1973], 3.3) | |
A reaction: This is nowadays accepted dogmatically as the biggest objection to behaviourism, but it could be challenged. Your analysis may begin by mentioning beliefs and desires, but if you keep going they may eventually fade out of the picture. |
3075 | Could a cloud have a headache if its particles formed into the right pattern? [Harman] |
Full Idea: If the right pattern of electrical discharges occurred in a cloud instead of in a brain, would that also be a headache? | |
From: Gilbert Harman (Thought [1973], 3.2) | |
A reaction: The standard objection to functionalism is to propose absurd implementations of a mind, but probably only a brain could produce the right electro-chemical combination. |
3086 | Are there any meanings apart from in a language? [Harman] |
Full Idea: The theory of language-independent meanings or semantic representations is mistaken. | |
From: Gilbert Harman (Thought [1973], 6.5) | |
A reaction: This would make him (in Dummett's terms) a 'philosopher of language' rather than a 'philosopher of thought'. Personally I disagree. Don't animals have 'meanings'? Can two sentences share a meaning? |
3078 | Speech acts, communication, representation and truth form a single theory [Harman] |
Full Idea: The various theories are not in competition. The theory of truth is part of the theory of representational character, which is presupposed by the theory of communication, which in turn is contained in the more general theory of speech acts. | |
From: Gilbert Harman (Thought [1973], 4.3) | |
A reaction: Certainly it seems that the supposed major contenders for a theory of meaning are just as much complements as they are competitors. |
3090 | There is only similarity in meaning, never sameness in meaning [Harman] |
Full Idea: The only sort of sameness of meaning we know is similarity in meaning, not exact sameness of meaning. | |
From: Gilbert Harman (Thought [1973], 6.8) | |
A reaction: The Eiffel Tower and le tour Eiffel? If you want to be difficult, you can doubt whether the word 'fast' ever has exactly the same meaning in two separate usages of the word. |
3082 | Ambiguity is when different underlying truth-conditional structures have the same surface form [Harman] |
Full Idea: Ambiguity results from the possibility of transforming different underlying truth-conditional structures into the same surface form. | |
From: Gilbert Harman (Thought [1973], 5.3) | |
A reaction: Personally I would call a 'truth-conditional structure' a 'proposition', and leave it to the philosophers to decide what a proposition is. |
3079 | Truth in a language is explained by how the structural elements of a sentence contribute to its truth conditions [Harman] |
Full Idea: A theory of truth for a language shows how the truth conditions of any sentence depend on the structure of that sentence. The theory will say, for each element of structure, what its contribution is. | |
From: Gilbert Harman (Thought [1973], 5.1) | |
A reaction: This just seems to push the problem of truth back a stage, as you need to know where the truth is to be found in the elements from which the structure is built. |
3085 | Sentences are different from propositions, since two sentences can express one proposition [Harman] |
Full Idea: 'Bob and John play golf' and 'John and Bob play golf' are equivalent; but if they were to be derived from the same underlying structure, one or the other of Bob and John would have to come first; and either possibility is arbitrary. | |
From: Gilbert Harman (Thought [1973], 6.4) | |
A reaction: If I watch Bob and John play golf, neither of them 'comes first'. A proposition about them need not involve 'coming first'. Only if you insist on formulating a sentence must you decide on that. |
3087 | The analytic/synthetic distinction is a silly division of thought into encyclopaedia and dictionary [Harman] |
Full Idea: No purpose is served by thinking that certain principles available to a person are contained in his internal encyclopaedia - and therefore only synthetic - whereas other principles are part of his internal dictionary - and are therefore analytic. | |
From: Gilbert Harman (Thought [1973], 6.5) | |
A reaction: If it led to two different ways to acquire knowledge, then quite a lot of purpose would be served. He speaks like a pragmatist. The question is whether some statements just are true because of some feature of meaning. Why not? |
3083 | Many predicates totally resist translation, so a universal underlying structure to languages is unlikely [Harman] |
Full Idea: There are many predicates of a given language that resist translation into another language, …so it is unlikely that there is a basic set of underlying structures common to all languages. | |
From: Gilbert Harman (Thought [1973], 5.4) | |
A reaction: Not convincing. 'Structures' are not the same as 'predicates'. Once a language has mapped its predicates, that blocks the intrusions of differently sliced alien predicates. No gaps. |