51 ideas
| 11147 | Naturalistic philosophers oppose analysis, preferring explanation to a priori intuition [Margolis/Laurence] |
| Full Idea: Philosophers who oppose conceptual analysis identify their approach as being 'naturalistic'. Philosophy is supposed to be continuous with science, and philosophical theories are to be defended on explanatory grounds, not by a priori intuitions. | |
| From: E Margolis/S Laurence (Concepts [2009], 5.2) | |
| A reaction: [They cite Papineau 1993, Devitt 1996 aand Kornblith 2002] I think there is a happy compromise here. I agree that any philosophical knowledge should be continuous with science, but we shouldn't prejudge how the analytic branch of science is done. |
| 18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
| Full Idea: Cohen's method of 'forcing' produces a new model of ZFC from an old model by appending a carefully chosen 'generic' set. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.4) |
| 18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
| Full Idea: A possible axiom is the Large Cardinal Axiom, which asserts that there are more and more stages in the cumulative hierarchy. Infinity can be seen as the first of these stages, and Replacement pushes further in this direction. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) |
| 18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
| Full Idea: The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
| Full Idea: In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| Full Idea: The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) |
| 18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
| Full Idea: A 'propositional function' is generated when one of the terms of the proposition is replaced by a variable, as in 'x is wise' or 'Socrates'. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: This implies that you can only have a propositional function if it is derived from a complete proposition. Note that the variable can be in either subject or in predicate position. It extends Frege's account of a concept as 'x is F'. |
| 18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
| Full Idea: The line of development that finally led to a coherent foundation for the calculus also led to the explicit introduction of completed infinities: each real number is identified with an infinite collection of rationals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.3) | |
| A reaction: Effectively, completed infinities just are the real numbers. |
| 18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
| Full Idea: Both Cantor's real number (Cauchy sequences of rationals) and Dedekind's cuts involved regarding infinite items (sequences or sets) as completed and subject to further manipulation, bringing the completed infinite into mathematics unambiguously. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1 n39) | |
| A reaction: So it is the arrival of the real numbers which is the culprit for lumbering us with weird completed infinites, which can then be the subject of addition, multiplication and exponentiation. Maybe this was a silly mistake? |
| 18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
| Full Idea: The stunning discovery that infinity comes in different degrees led to the theory of infinite cardinal numbers, the heart of contemporary set theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: It occurs to me that these huge cardinals only exist in set theory. If you took away that prop, they would vanish in a puff. |
| 18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
| Full Idea: By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion. |
| 18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
| Full Idea: An 'inaccessible' cardinal is one that cannot be reached by taking unions of small collections of smaller sets or by taking power sets. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.5) | |
| A reaction: They were introduced by Hausdorff in 1908. |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| Full Idea: Even the fundamental theorems about limits could not [at first] be proved because the reals themselves were not well understood. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This refers to the period of about 1850 (Weierstrass) to 1880 (Dedekind and Cantor). |
| 18182 | The extension of concepts is not important to me [Maddy] |
| Full Idea: I attach no decisive importance even to bringing in the extension of the concepts at all. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], §107) | |
| A reaction: He almost seems to equate the concept with its extension, but that seems to raise all sorts of questions, about indeterminate and fluctuating extensions. |
| 18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
| Full Idea: In the ZFC cumulative hierarchy, Frege's candidates for numbers do not exist. For example, new three-element sets are formed at every stage, so there is no stage at which the set of all three-element sets could he formed. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Ah. This is a very important fact indeed if you are trying to understand contemporary discussions in philosophy of mathematics. |
| 18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
| Full Idea: To solve the Julius Caesar problem, Frege requires explicit definitions of the numbers, and he proposes his well-known solution: the number of Fs = the extension of the concept 'equinumerous with F' (based on one-one correspondence). | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Why do there have to be Fs before there can be the corresponding number? If there were no F for 523, would that mean that '523' didn't exist (even if 522 and 524 did exist)? |
| 18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
| Full Idea: The set theory axioms developed in producing foundations for mathematics also have strong consequences for existing fields, and produce a theory that is immensely fruitful in its own right. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: [compressed] Second of Maddy's three benefits of set theory. This benefit is more questionable than the first, because the axioms may be invented because of their nice fruit, instead of their accurate account of foundations. |
| 18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
| Full Idea: The single unified area of set theory provides a court of final appeal for questions of mathematical existence and proof. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Maddy's third benefit of set theory. 'Existence' means being modellable in sets, and 'proof' means being derivable from the axioms. The slightly ad hoc character of the axioms makes this a weaker defence. |
| 18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
| Full Idea: Set theoretic foundations bring all mathematical objects and structures into one arena, allowing relations and interactions between them to be clearly displayed and investigated. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: The first of three benefits of set theory which Maddy lists. The advantages of the one arena seem to be indisputable. |
| 18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
| Full Idea: The identification of geometric points with real numbers was among the first and most dramatic examples of the power of set theoretic foundations. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: Hence the clear definition of the reals by Dedekind and Cantor was the real trigger for launching set theory. |
| 18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
| Full Idea: The structure of a geometric line by rational points left gaps, which were inconsistent with a continuous line. Set theory provided an ordering that contained no gaps. These reals are constructed from rationals, which come from integers and naturals. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.2) | |
| A reaction: This completes the reduction of geometry to arithmetic and algebra, which was launch 250 years earlier by Descartes. |
| 18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
| Full Idea: Our much loved mathematical knowledge rests on two supports: inexorable deductive logic (the stuff of proof), and the set theoretic axioms. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I Intro) |
| 18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
| Full Idea: It could turn out that all applications of continuum mathematics in natural sciences are actually instances of idealisation. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) |
| 18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
| Full Idea: Crudely, the scientist posits only those entities without which she cannot account for observations, while the set theorist posits as many entities as she can, short of inconsistency. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.5) |
| 18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
| Full Idea: Recent commentators have noted that Frege's versions of the basic propositions of arithmetic can be derived from Hume's Principle alone, that the fatal Law V is only needed to derive Hume's Principle itself from the definition of number. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], I.1) | |
| A reaction: Crispin Wright is the famous exponent of this modern view. Apparently Charles Parsons (1965) first floated the idea. |
| 18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
| Full Idea: The case of atoms makes it clear that the indispensable appearance of an entity in our best scientific theory is not generally enough to convince scientists that it is real. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: She refers to the period between Dalton and Einstein, when theories were full of atoms, but there was strong reluctance to actually say that they existed, until the direct evidence was incontrovertable. Nice point. |
| 11141 | Modern empiricism tends to emphasise psychological connections, not semantic relations [Margolis/Laurence] |
| Full Idea: A growing number of philosophers are attracted to modified forms of empiricism, emphasizing psychological relations between the conceptual system and perceptual and motor states, not semantic relations. | |
| From: E Margolis/S Laurence (Concepts [2009], 3.2) | |
| A reaction: I suddenly spot that this is what I have been drifting towards for some time! The focus is concept formation, where the philosophers need to join forces with the cognitive scientists. |
| 18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
| Full Idea: In science we treat the earth's surface as flat, we assume the ocean to be infinitely deep, we use continuous functions for what we know to be quantised, and we take liquids to be continuous despite atomic theory. | |
| From: Penelope Maddy (Naturalism in Mathematics [1997], II.6) | |
| A reaction: If fussy people like scientists do this all the time, how much more so must the confused multitude be doing the same thing all day? |
| 11142 | Body-type seems to affect a mind's cognition and conceptual scheme [Margolis/Laurence] |
| Full Idea: It is claimed, on the basis of empirical research, that the type of body that an organism has profoundly affects it cognitive operations and the way it conceptualises the world. We can't assume that human minds could inhere in wildly different body types. | |
| From: E Margolis/S Laurence (Concepts [2009], 3.2) | |
| A reaction: Sounds interesting. They cite Lawrence Shapiro 2004. It needs a large effort of imagination to think how a snake or whale or albatross might conceptualise the world, in relation to their bodies. |
| 11121 | Language of thought has subject/predicate form and includes logical devices [Margolis/Laurence] |
| Full Idea: The language of thought is taken to have subject/predicate form and include logical devices, such as quantifiers and variables. | |
| From: E Margolis/S Laurence (Concepts [2009], 1.1) |
| 11120 | Concepts are either representations, or abilities, or Fregean senses [Margolis/Laurence] |
| Full Idea: The three main options for the ontological status of concepts are to identify them with mental representations, or with abilities, or with Fregean senses. | |
| From: E Margolis/S Laurence (Concepts [2009], 1) |
| 11122 | A computer may have propositional attitudes without representations [Margolis/Laurence] |
| Full Idea: It may be possible to have propositional attitudes without having the mental representations tokened in one's head. ...We may say a chess-playing computer thinks it should develop its queen early, though we know it has no representation with that content. | |
| From: E Margolis/S Laurence (Concepts [2009], 1.1) | |
| A reaction: [Thye cite Dennett - who talks of the 'intentional stance'] It is, of course, a moot point whether we would attribute a propositional attitude (such as belief) to a machine once we knew that it wasn't representing the relevant concepts. |
| 11124 | Do mental representations just lead to a vicious regress of explanations [Margolis/Laurence] |
| Full Idea: A standard criticism is that the mental representation view of concepts creates just another item whose significance bears explaining. Either we have a vicious regress, or we might as well explain external language directly. | |
| From: E Margolis/S Laurence (Concepts [2009], 1.2) | |
| A reaction: [They cite Dummett, with Wittgenstein in the background] I don't agree, because I think that explanation of concepts only stops when it dovetails into biology. |
| 11123 | Maybe the concept CAT is just the ability to discriminate and infer about cats [Margolis/Laurence] |
| Full Idea: The view that concepts are abilities (e.g. found in Brandom, Dummett and Millikan) would say that the concept CAT amounts to the ability to discriminate cats from non-cats and to draw certain inferences about cats. | |
| From: E Margolis/S Laurence (Concepts [2009], 1.2) | |
| A reaction: Feels wrong. The concept is what makes these abilities possible, but it seems rather behaviourist to identify the concept with what is enabled by the concept. You might understand 'cat', but fail to recognise your first cat (though you might suspect it). |
| 11125 | The abilities view cannot explain the productivity of thought, or mental processes [Margolis/Laurence] |
| Full Idea: The abilities view of concepts, by its rejection of mental representation, is ill-equipped to explain the productivity of thought; and it can say little about mental processes. | |
| From: E Margolis/S Laurence (Concepts [2009], 1.2) | |
| A reaction: The latter point arises from its behaviouristic character, which just gives us a black box with some output of abilities. In avoiding a possible regress, it offers no explanation at all. |
| 11140 | Concept-structure explains typicality, categories, development, reference and composition [Margolis/Laurence] |
| Full Idea: The structures of concepts are invoked to explain typicality effects, reflective categorization, cognitive development, reference determination, and compositionality. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.5) |
| 11128 | Classically, concepts give necessary and sufficient conditions for falling under them [Margolis/Laurence] |
| Full Idea: The classical theory is that a concept has a definitional structure in that it is composed of simpler concepts that express necessary and sufficient conditions for falling under the concept, the stock example being unmarried and a man for 'bachelor'. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.1) | |
| A reaction: This is the background idea to philosophy as analysis, and it makes concepts essentially referential, in that they are defined by their ability to pick things out. There must be some degree of truth in the theory. |
| 11130 | Typicality challenges the classical view; we see better fruit-prototypes in apples than in plums [Margolis/Laurence] |
| Full Idea: The classical view is challenged by the discovery that certain categories are taken to be more typical, with typicality widely correlating with other data. Apples are judged to be more typical of (and have more common features with) fruit than plums are. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.1) | |
| A reaction: This discovery that people use prototypes in thinking has been the biggest idea to ever hit the philosophy of concepts, and simply cannot be ignored (as long as the research keeps reinforcing it, which I believe it does). The classical view might adapt. |
| 11129 | The classical theory explains acquisition, categorization and reference [Margolis/Laurence] |
| Full Idea: The appeal of the classical theory of concepts is that it offers unified treatments of concept acquisition (assembling constituents), categorization (check constituents against target), and reference determination (whether they apply). | |
| From: E Margolis/S Laurence (Concepts [2009], 2.1) | |
| A reaction: [See Idea 11128 for the theory] As so often, I find myself in sympathy with the traditional view which has been relegated to ignominy by our wonderful modern philosophers. |
| 11131 | It may be that our concepts (such as 'knowledge') have no definitional structure [Margolis/Laurence] |
| Full Idea: In the light of problems such as the definition of knowledge, many philosophers now take seriously the possibility that our concepts lack definitional structure. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.1) | |
| A reaction: This challenges the classical view, that there are precise conditions for each concept. That view would obviously be in difficulties with atomic concepts, so our account of those might be applied all the way up. |
| 11132 | The prototype theory is probabilistic, picking something out if it has sufficient of the properties [Margolis/Laurence] |
| Full Idea: In the prototype theory of concepts, a lexical concept has probabilistic structure in that something falls under it if it satisfies a sufficient number of properties encoded by the constituents. It originates in Wittgenstein's 'family resemblance'. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.2) | |
| A reaction: It would seem unlikely to be a matter of the 'number' of properties, and would have to involve some notion of what was essential to the prototype. |
| 11133 | Prototype theory categorises by computing the number of shared constituents [Margolis/Laurence] |
| Full Idea: On the prototype theory, categorization is to be understood as a similarity comparison process, where similarity is computed as a function of the number of constituents that two concepts hold in common. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.2) | |
| A reaction: Again it strikes me that 'computing' similarity by mere 'number' of shared constituents won't do, as there is a prior judgement about which constituents really matter, or are essential. That may even be hard-wired. |
| 11134 | People don't just categorise by apparent similarities [Margolis/Laurence] |
| Full Idea: When it comes to more reflexive judgements, people go beyond the outcome of a similarity comparison. Even children say that a dog surgically altered to look like a raccoon is still a dog. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.2) | |
| A reaction: We can defend the theory by not underestimating people so much. Most categorisation is done on superficial grounds, but even children know there may be hidden similarities (behind the mask, under the bonnet) which are more important. |
| 11135 | Complex concepts have emergent properties not in the ingredient prototypes [Margolis/Laurence] |
| Full Idea: An objection to the prototype view concerns compositionality. A complex concept often has emergent properties, as when it seems that 'pet fish' encodes for brightly coloured, which has no basis in the prototypes for 'pet' or 'fish'. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.2) | |
| A reaction: I would take 'pet fish' to work like a database query. 'Fish' has a very vague prototype, and then 'pet fish' narrows the search to fish which are appropriate to be pets. We might say that the prototype is refined, or the Mk 2 prototype appears. |
| 11136 | Many complex concepts obviously have no prototype [Margolis/Laurence] |
| Full Idea: Many patently complex concepts don't even have a prototype structure, such as 'Chairs that were purchased on a Wednesday'. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.2) | |
| A reaction: [The example seems to be from Fodor] I disagree. If we accept the notion of 'refining' the prototype (see Idea 11135), then the compositionality of the expression will produce a genuine but very unusual prototype. |
| 11137 | The theory theory of concepts says they are parts of theories, defined by their roles [Margolis/Laurence] |
| Full Idea: The theory theory of concepts says that terms are related as in a scientific theory, and that categorization resembles theorising. It is generally assumed that scientific terms are interdefined so that content is determined by its role in the theory. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.3) | |
| A reaction: I never like this sort of account. What are the characteristics of the thing which enable it to fulfil its role? You haven't defined a car when you've said it gets you from A to B. |
| 11138 | The theory theory is holistic, so how can people have identical concepts? [Margolis/Laurence] |
| Full Idea: A problem with the theory theory of concepts is that it is holistic, saying a concept is determined by its role, not by its constituents. It then seems difficult for different people to possess the same concepts (or even the same person, over time). | |
| From: E Margolis/S Laurence (Concepts [2009], 2.3) | |
| A reaction: This seems a good objection to any holistic account of concepts or meaning - spotted by Plato in motivating his theory of Forms, to give the necessary stability to communication. |
| 11139 | Maybe concepts have no structure, and determined by relations to the world, not to other concepts [Margolis/Laurence] |
| Full Idea: According to conceptual atomism, lexical concepts have no semantic structure, and the content of a concept isn't determined by its relation to other concepts but by its relations to the world. | |
| From: E Margolis/S Laurence (Concepts [2009], 2.4) | |
| A reaction: [They cite Fodor 1998 and Millikan 2000] I like the sound of that, because I take the creation of concepts to be (in the first instance) a response to the world, not a response to other concepts. |
| 11146 | People can formulate new concepts which are only named later [Margolis/Laurence] |
| Full Idea: People seem to be able to formulate novel concepts which are left to be named later; the concept comes first, the name second. | |
| From: E Margolis/S Laurence (Concepts [2009], 4.2) | |
| A reaction: [This seems to have empirical support, and he cites Pinker 1994] I do not find this remotely surprising, since I presume that human concepts are a continuous kind with animal concepts, including non-conscious concepts (why not?). |
| 7861 | Libet says the processes initiated in the cortex can still be consciously changed [Libet, by Papineau] |
| Full Idea: Libet himself points out that the conscious decisions still have the power to 'endorse' or 'cancel', so to speak, the processes initiated by the earlier cortical activity: no action will result if the action's execution is consciously countermanded. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by David Papineau - Thinking about Consciousness 1.4 | |
| A reaction: This is why Libet's findings do not imply 'epiphenomenalism'. It seems that part of a decisive action is non-conscious, undermining the all-or-nothing view of consciousness. Searle tries to smuggle in free will at this point (Idea 3817). |
| 6660 | Libet found conscious choice 0.2 secs before movement, well after unconscious 'readiness potential' [Libet, by Lowe] |
| Full Idea: Libet found that a subject's conscious choice to move was about a fifth of a second before movement, and thus later than the onset of the brain's so-called 'readiness potential', which seems to imply that unconscious processes initiates action. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.9 | |
| A reaction: Of great interest to philosophers! It seems to make conscious choices epiphenomenal. The key move, I think, is to give up the idea of consciousness as being all-or-nothing. My actions are still initiated by 'me', but 'me' shades off into unconsciousness. |