28 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 4986 | A weaker kind of reductionism than direct translation is the use of 'bridge laws' [Kirk,R] |
| Full Idea: If multiple realisability means that psychological terms cannot be translated into physics, one weaker kind of reductionism resorts to 'bridge laws' which link the theory to be reduced to the reducing theory. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: It seems to me that reduction is all-or-nothing, so there can't be a 'weaker' kind. If they are totally separate but linked by naturally necessary laws (e.g. low temperature and ice), they are supervenient, but not reducible to one another. |
| 6493 | We are not conscious of pure liquidity, but of the liquidity of water [Firth] |
| Full Idea: We are not conscious of liquidity, coldness, and solidity, but of the liquidity of water, the coldness of ice, and the solidity of rocks. | |
| From: Roderick Firth (Sense Data and the Percept Theory [1949]), quoted by Howard Robinson - Perception 1.7 | |
| A reaction: A nice point, but it might not be entirely true in a blindfold test, where one might only report properties like 'sticky' or 'warm', without having any clear concept of the substance being experienced. Firth is proposing the 'percept theory'. |
| 5001 | Maybe we should see intentionality and consciousness as a single problem, not two [Kirk,R] |
| Full Idea: Many philosophers today have adopted the view that we can achieve an enormous simplification by reducing the two components of the mind-body problem - intentionality and consciousness - into one; ...consciousness is no more than representations. | |
| From: Robert Kirk (Mind and Body [2003], §8.4) | |
| A reaction: One would then see subjective experience and informational content as two consequences of a single mental activity. This strikes me as the correct route to go. We do, after all, learn BY experiencing. Hence concepts are tied in with qualia. |
| 4993 | If a bird captures a worm, we could say its behaviour is 'about' the worm [Kirk,R] |
| Full Idea: When a bird pulls a worm from the ground, then swallows it piece by piece, there is a sense in which its behaviour can be said to be about the worm. | |
| From: Robert Kirk (Mind and Body [2003], §5.4) | |
| A reaction: This is preparing the ground for a possible behaviourist account of intentionality. Reply: you could say rain is about puddles, or you could say we have adopted Dennett's 'intentional stance' to birds, but it tells us nothing about their psychology. |
| 5000 | Behaviourism says intentionality is an external relation; language of thought says it's internal [Kirk,R] |
| Full Idea: The conflict over whether intentionality is a matter of behavioural relations with the rest of the world, or of the internal states of the subject, is at its most dramatic in the contrast between behaviourism and the language of thought hypothesis. | |
| From: Robert Kirk (Mind and Body [2003], §7.10) | |
| A reaction: I just don't believe any behaviourist external account of intentionality, which ducks the question of how it all works. Personally I am more drawn to maps and models than to a language of thought. I plan my actions in an imagined space-time world. |
| 4982 | Dualism implies some brain events with no physical cause, and others with no physical effect [Kirk,R] |
| Full Idea: If the mind causes brain events, then they are not caused by other brain events, and such causal gaps should be detectable by scientists; there should also be a gap of brain-events which cause no other brain events, because they are causing mind events. | |
| From: Robert Kirk (Mind and Body [2003], §2.5) | |
| A reaction: This is the double causation problem which Spinoza had spotted (Idea 4862). Expressed this way, it seems a screamingly large problem for dualism. We should be able to discover some VERY strange physical activity in the brain. |
| 4991 | Behaviourism seems a good theory for intentional states, but bad for phenomenal ones [Kirk,R] |
| Full Idea: For many kinds of mental states, notably intentional ones such as beliefs and desires, behaviourism is appealing, ..but for sensations and experiences such as pain, it seems grossly implausible. | |
| From: Robert Kirk (Mind and Body [2003], §5.1) | |
| A reaction: The theory does indeed make a bit more sense for intentional states, but it still strikes me as nonsense that there is no more to my belief that 'Whales live in the Atlantic' than a disposition to say something. WHY do I say this something? |
| 4994 | Behaviourism offers a good alternative to simplistic unitary accounts of mental relationships [Kirk,R] |
| Full Idea: There is a temptation to think that 'aboutness', and the 'contents' of thoughts, and the relation of 'reference', are single and unitary relationships, but behaviourism offers an alternative approach. | |
| From: Robert Kirk (Mind and Body [2003], §5.5) | |
| A reaction: Personally I wouldn't touch behaviourism with a barge-pole (as it ducks the question of WHY certain behaviour occurs), but a warning against simplistic accounts of intentional states is good. I am sure there cannot be a single neat theory of refererence. |
| 4992 | In 'holistic' behaviourism we say a mental state is a complex of many dispositions [Kirk,R] |
| Full Idea: There is a non-reductive version of behaviourism ( which we can call 'global' or 'holistic') which says there is no more to having mental states than having a complex of certain kinds of behavioural dispositions. | |
| From: Robert Kirk (Mind and Body [2003], §5.2) | |
| A reaction: This is designed to meet a standard objection to behaviourism - that there is no straight correlation between what I think and how I behave. The present theory is obviously untestable, because a full 'complex' of human dispositions is never repeated. |
| 4990 | The inverted spectrum idea is often regarded as an objection to behaviourism [Kirk,R] |
| Full Idea: The inverted spectrum idea is often regarded as an objection to behaviourism. | |
| From: Robert Kirk (Mind and Body [2003], §4.5) | |
| A reaction: Thus, my behaviour at traffic lights should be identical, even if I have a lifelong inversion of red and green. A good objection. Note that physicalists can believe in inverted qualia as well a dualists, as long as the brain states are also inverted. |
| 4984 | All meaningful psychological statements can be translated into physics [Kirk,R] |
| Full Idea: All psychological statements which are meaningful, that is to say, which are in principle verifiable, are translatable into propositions which do not involve psychological concepts, but only the concepts of physics. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: This shows how eliminativist behaviourism arises out of logical positivism (by only allowing what is verifiable). The simplest objection: we can't verify the mental states of others, because they are private, but they are still the best explanation. |
| 4998 | Instead of representation by sentences, it can be by a distribution of connectionist strengths [Kirk,R] |
| Full Idea: In a connectionist system, information is represented not by sentences but by the total distribution of connection strengths. | |
| From: Robert Kirk (Mind and Body [2003], §7.6) | |
| A reaction: Neither sentences (of a language of thought) NOR connection strengths strike me as very plausible ways for a brain to represent things. It must be something to do with connections, but it must also be to do with neurons, or we get bizarre counterexamples. |
| 4985 | If mental states are multiply realisable, they could not be translated into physical terms [Kirk,R] |
| Full Idea: If psychological states are multiply realisable it is hard to see how they could possibly be translated into physical terms. | |
| From: Robert Kirk (Mind and Body [2003], §3.8) | |
| A reaction: Reductive funtionalism would do it. A writing iimplement is physical and multiply realisable. Personally I prefer the strategy of saying mental states are NOT multiply realisable. If frog brains differ from ours, they probably don't feel pain like us. |
| 4997 | It seems unlikely that most concepts are innate, if a theory must be understood to grasp them [Kirk,R] |
| Full Idea: It is widely accepted that for many concepts, if not all, grasping the concept requires grasping some theory, ...which makes difficulties for the view that concepts are not learned: for 'radical concept nativism', as Fodor calls it. | |
| From: Robert Kirk (Mind and Body [2003], §7.3) | |
| A reaction: Not a problem for traditional rationalist theories, where the whole theory can be innate along with the concept, but a big objection to modern more cautious non-holistic views (such as Fodor's). Does a bird have a concept AND theory of a nest? |
| 4999 | For behaviourists language is just a special kind of behaviour [Kirk,R] |
| Full Idea: Behaviourists regard the use of language as just a special kind of behaviour. | |
| From: Robert Kirk (Mind and Body [2003], §7.9) | |
| A reaction: This is not an intuitively obvious view of language. We behave, and then we talk about behaviour. Performative utterances (like promising) have an obvious behavioural aspect, as do violent threats, but not highly theoretical language (such as maths). |
| 4995 | Behaviourists doubt whether reference is a single type of relation [Kirk,R] |
| Full Idea: To most behaviourists it seems misguided to expect there to be a single relation that connects referring expressions with their referents. | |
| From: Robert Kirk (Mind and Body [2003], §5.5) | |
| A reaction: You don't need to be a behaviourist to feel this doubt. Think about names of real people, names of fictional people, reference to misunderstood items, or imagined items, or reference in dreams, or to mathematical objects, or negations etc. |