97 ideas
| 18559 | Philosophy is empty if it does not in some way depend on matters of fact [Machery] |
| Full Idea: Save, maybe, for purely formal (e.g. logical) theories, philosophical claims whose correctness does not depend, however indirectly, on matters of fact are empty: they are neither true nor false. | |
| From: Edouard Machery (Doing Without Concepts [2009], Intro) | |
| A reaction: I subscribe to this view. I'd even say that logic is empty if it is not answerable to the facts. The facts are nature, so this is a naturalistic manifesto. |
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| Full Idea: The notion of a function evolved gradually from wanting to see what curves can be represented as trigonometric series. The study of arbitrary functions led Cantor to the ordinal numbers, which led to set theory. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| Full Idea: Cantor's Theorem says that for any set x, its power set P(x) has more members than x. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| Full Idea: Cantor's diagonalisation argument generalises to show that any set has more subsets than it has members. | |
| From: report of George Cantor (works [1880]) by David Bostock - Philosophy of Mathematics 4.5 | |
| A reaction: Thus three members will generate seven subsets. This means that 'there is no end to the series of cardinal numbers' (Bostock p.106). |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| Full Idea: Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one. | |
| From: report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1 | |
| A reaction: There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it. |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| Full Idea: Cantor's theories exhibited the contradictions others had claimed to derive from the supposition of infinite sets as confusions resulting from the failure to mark the necessary distinctions with sufficient clarity. | |
| From: report of George Cantor (works [1880]) by Michael Potter - Set Theory and Its Philosophy Intro 1 |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| Full Idea: Cantor discovered that the continuum is the powerset of the integers. While adding or multiplying infinities didn't move up a level of complexity, multiplying a number by itself an infinite number of times did. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| Full Idea: Cantor first stated the Union Axiom in a letter to Dedekind in 1899. It is nearly too obvious to deserve comment from most commentators. Justifications usually rest on 'limitation of size' or on the 'iterative conception'. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Believing the Axioms I §1.3 | |
| A reaction: Surely someone can think of some way to challenge it! An opportunity to become notorious, and get invited to conferences. |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack. | |
| From: report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro | |
| A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets. |
| 17896 | We need to know the meaning of 'and', prior to its role in reasoning [Prior,AN, by Belnap] |
| Full Idea: For Prior, so the moral goes, we must first have a notion of what 'and' means, independently of the role it plays as premise and as conclusion. | |
| From: report of Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.132 | |
| A reaction: The meaning would be given by the truth tables (the truth-conditions), whereas the role would be given by the natural deduction introduction and elimination rules. This seems to be the basic debate about logical connectives. |
| 17898 | Prior's 'tonk' is inconsistent, since it allows the non-conservative inference A |- B [Belnap on Prior,AN] |
| Full Idea: Prior's definition of 'tonk' is inconsistent. It gives us an extension of our original characterisation of deducibility which is not conservative, since in the extension (but not the original) we have, for arbitrary A and B, A |- B. | |
| From: comment on Arthur N. Prior (The Runabout Inference Ticket [1960]) by Nuel D. Belnap - Tonk, Plonk and Plink p.135 | |
| A reaction: Belnap's idea is that connectives don't just rest on their rules, but also on the going concern of normal deduction. |
| 11021 | Prior rejected accounts of logical connectives by inference pattern, with 'tonk' his absurd example [Prior,AN, by Read] |
| Full Idea: Prior dislike the holism inherent in the claim that the meaning of a logical connective was determined by the inference patterns into which it validly fitted. ...His notorious example of 'tonk' (A → A-tonk-B → B) was a reductio of the view. | |
| From: report of Arthur N. Prior (The Runabout Inference Ticket [1960]) by Stephen Read - Thinking About Logic Ch.8 | |
| A reaction: [The view being attacked was attributed to Gentzen] |
| 13836 | Maybe introducing or defining logical connectives by rules of inference leads to absurdity [Prior,AN, by Hacking] |
| Full Idea: Prior intended 'tonk' (a connective which leads to absurdity) as a criticism of the very idea of introducing or defining logical connectives by rules of inference. | |
| From: report of Arthur N. Prior (The Runabout Inference Ticket [1960], §09) by Ian Hacking - What is Logic? |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'. | |
| From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3 | |
| A reaction: [Smith summarises the diagonal argument] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does. |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly? |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| Full Idea: Cantor believed he had discovered that between the finite and the 'Absolute', which is 'incomprehensible to the human understanding', there is a third category, which he called 'the transfinite'. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.4 |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| Full Idea: In 1878 Cantor published the unexpected result that one can put the points on a plane, or indeed any n-dimensional space, into one-to-one correspondence with the points on a line. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| Full Idea: Cantor took the ordinal numbers to be primary: in his generalization of the cardinals and ordinals into the transfinite, it is the ordinals that he calls 'numbers'. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind VI | |
| A reaction: [Tait says Dedekind also favours the ordinals] It is unclear how the matter might be settled. Humans cannot give the cardinality of large groups without counting up through the ordinals. A cardinal gets its meaning from its place in the ordinals? |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| Full Idea: Cantor taught us to regard the totality of natural numbers, which was formerly thought to be infinite, as really finite after all. | |
| From: report of George Cantor (works [1880]) by John Mayberry - What Required for Foundation for Maths? p.414-2 | |
| A reaction: I presume this is because they are (by definition) countable. |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| Full Idea: Cantor introduced the distinction between cardinal and ordinal numbers. | |
| From: report of George Cantor (works [1880]) by William W. Tait - Frege versus Cantor and Dedekind Intro | |
| A reaction: This seems remarkably late for what looks like a very significant clarification. The two concepts coincide in finite cases, but come apart in infinite cases (Tait p.58). |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| Full Idea: Cantor's work revealed that the notion of an ordinal number is more fundamental than that of a cardinal number. | |
| From: report of George Cantor (works [1880]) by Michael Dummett - Frege philosophy of mathematics Ch.23 | |
| A reaction: Dummett makes it sound like a proof, which I find hard to believe. Is the notion that I have 'more' sheep than you logically prior to how many sheep we have? If I have one more, that implies the next number, whatever that number may be. Hm. |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| Full Idea: The cardinal number of M is the general idea which, by means of our active faculty of thought, is deduced from the collection M, by abstracting from the nature of its diverse elements and from the order in which they are given. | |
| From: George Cantor (works [1880]), quoted by Bertrand Russell - The Principles of Mathematics §284 | |
| A reaction: [Russell cites 'Math. Annalen, XLVI, §1'] See Fine 1998 on this. |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| Full Idea: Cantor said he could show that every infinite set of points on the line could be placed into one-to-one correspondence with either the natural numbers or the real numbers - with no intermediate possibilies (the Continuum hypothesis). His proof failed. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.1 |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| Full Idea: Cantor's diagonal argument showed that all the infinite decimals between 0 and 1 cannot be written down even in a single never-ending list. | |
| From: report of George Cantor (works [1880]) by Stephen Read - Thinking About Logic Ch.6 |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| Full Idea: Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6 | |
| A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number. |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2 |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| Full Idea: From the very nature of an irrational number, it seems necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. Infinite classes are obvious in the Dedekind Cut, but have logical difficulties | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II Intro | |
| A reaction: Almost the whole theory of analysis (calculus) rested on the irrationals, so a theory of the infinite was suddenly (in the 1870s) vital for mathematics. Cantor wasn't just being eccentric or mystical. |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| Full Idea: Cantor's 1891 diagonal argument revealed there are infinitely many infinite powers. Indeed, it showed more: it shows that given any set there is another of greater power. Hence there is an infinite power strictly greater than that of the set of the reals. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite III.2 |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| Full Idea: What we might call 'Cantor's Thesis' is that there won't be a potential infinity of any sort unless there is an actual infinity of some sort. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: This idea is nicely calculated to stop Aristotle in his tracks. |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| Full Idea: Cantor showed that the complete totality of natural numbers cannot be mapped 1-1 onto the complete totality of the real numbers - so there are different sizes of infinity. | |
| From: report of George Cantor (works [1880]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.4 |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| Full Idea: Cantor's 'continuum hypothesis' is the assertion that there are no infinite cardinalities strictly between the size of the natural numbers and the size of the real numbers. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Thinking About Mathematics 2.4 | |
| A reaction: The tricky question is whether this hypothesis can be proved. |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| Full Idea: Cantor's Continuum Hypothesis (CH) says that for every infinite set X of reals there is either a one-to-one correspondence between X and the natural numbers, or between X and the real numbers. | |
| From: report of George Cantor (works [1880]) by Peter Koellner - On the Question of Absolute Undecidability 1.2 | |
| A reaction: Every single writer I read defines this differently, which drives me crazy, but is also helpfully illuminating. There is a moral there somewhere. |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| Full Idea: Cantor conjectured that there is no size between those of the naturals and the reals - called the 'continuum hypothesis'. The generalized version says that for no infinite set A is there a set larger than A but smaller than P(A). | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: Thus there are gaps between infinite numbers, and the power set is the next size up from any infinity. Much discussion as ensued about whether these two can be proved. |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| Full Idea: Cantor's Continuum Hypothesis states that there are no sets which are too large for there to be a one-to-one correspondence between the set and the natural numbers, but too small for there to exist a one-to-one correspondence with the real numbers. | |
| From: report of George Cantor (works [1880]) by Leon Horsten - Philosophy of Mathematics §5.1 |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| Full Idea: Cantor's conjecture (the Continuum Hypothesis) is that there are no sets between N and P(N). The 'generalized' version replaces N with an arbitrary infinite set. | |
| From: report of George Cantor (works [1880]) by Robert S. Wolf - A Tour through Mathematical Logic 2.2 | |
| A reaction: The initial impression is that there is a single gap in the numbers, like a hole in ozone layer, but the generalised version implies an infinity of gaps. How can there be gaps in the numbers? Weird. |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| Full Idea: Cantor's Continuum Hypothesis was that there is no cardinal number greater than aleph-null but less than the cardinality of the continuum. | |
| From: report of George Cantor (works [1880]) by Charles Chihara - A Structural Account of Mathematics 05.1 | |
| A reaction: I have no view on this (have you?), but the proposal that there are gaps in the number sequences has to excite all philosophers. |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| Full Idea: Cantor's set theory was not of collections in some familiar sense, but of collections that can be counted using the indexes - the finite and transfinite ordinal numbers. ..He treated infinite collections as if they were finite. | |
| From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite I |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| Full Idea: Cantor's second innovation was to extend the sequence of ordinal numbers into the transfinite, forming a handy scale for measuring infinite cardinalities. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: Struggling with this. The ordinals seem to locate the cardinals, but in what sense do they 'measure' them? |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| Full Idea: Cantor's first innovation was to treat cardinality as strictly a matter of one-to-one correspondence, so that the question of whether two infinite sets are or aren't of the same size suddenly makes sense. | |
| From: report of George Cantor (works [1880]) by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: It makes sense, except that all sets which are infinite but countable can be put into one-to-one correspondence with one another. What's that all about, then? |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| Full Idea: Cantor's theorem entails that there are more property extensions than objects. So there are not enough objects in any domain to serve as extensions for that domain. So Frege's view that numbers are objects led to the Caesar problem. | |
| From: report of George Cantor (works [1880]) by Stewart Shapiro - Philosophy of Mathematics 4.6 | |
| A reaction: So the possibility that Caesar might have to be a number arises because otherwise we are threatening to run out of numbers? Is that really the problem? |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| Full Idea: Pure mathematics ...according to my conception is nothing other than pure set theory. | |
| From: George Cantor (works [1880], I.1), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
| A reaction: [an unpublished paper of 1884] So right at the beginning of set theory this claim was being made, before it was axiomatised, and so on. Zermelo endorsed the view, and it flourished unchallenged until Benacerraf (1965). |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 18564 | Do categories store causal knowledge, or typical properties, or knowledge of individuals? [Machery] |
| Full Idea: Psychologists have attempted to determine whether a concept of a category stores some causal knowledge about the members, some knowledge about their typical properties, or some knowledge about specific members. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.3.2) | |
| A reaction: I take there to be a psychological process of 'generalisation', so that knowledge of individuals is not and need not be retained. I am dubious about entities called 'properties', so I will vote for causal (including perceptual) knowledge. |
| 18604 | Are quick and slow categorisation the same process, or quite different? [Machery] |
| Full Idea: Are categorisation under time pressure and categorisation without time pressure ...two different cognitive competences? | |
| From: Edouard Machery (Doing Without Concepts [2009], 5.1.1) | |
| A reaction: This is a psychologist's question. Introspectively, they do seem to be rather different, as there is no time for theorising and explaining when you are just casting your eyes over the landscape. |
| 18573 | For each category of objects (such as 'dog') an individual seems to have several concepts [Machery] |
| Full Idea: I contend that the best available evidence suggests that for each category of objects an individual typically has several concepts. For instance, instead of having a single concept of dog, an individual has in fact several concepts of dog. | |
| From: Edouard Machery (Doing Without Concepts [2009], 3) | |
| A reaction: Machery's book is a sustained defence of this hypothesis, with lots of examples from psychology. Any attempt by philosophers to give a neat and tidy account of categorisation looks doomed. |
| 18602 | A thing is classified if its features are likely to be generated by that category's causal laws [Machery] |
| Full Idea: A to-be-classified object is considered a category member to the extent that its features were likely to have been generated by the category's causal laws. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.4.4) | |
| A reaction: [from Bob Rehder, psychologist, 2003] This is an account of categorisation which arises from the Theory Theory view of concepts, of which I am a fan. I love this idea, which slots neatly into the account I have been defending. Locke would like this. |
| 18565 | There may be ad hoc categories, such as the things to pack in your suitcase for a trip [Machery] |
| Full Idea: There may be ad hoc categories, as when people think about the things to pack in a small suitcase for a trip abroad. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.4.1) | |
| A reaction: This seems to be obviously correct, though critics might say that 'category' is too grand a term for such a grouping. |
| 18570 | There may be several ways to individuate things like concepts [Machery] |
| Full Idea: Philosophers have rarely explained why they believe that there is a single correct way of individuating concepts. Many entities can be legitimately individuated in several ways. | |
| From: Edouard Machery (Doing Without Concepts [2009], 2.1.3) | |
| A reaction: I cite this under 'individuation' because I think that is a very garbled concept. I agree with this point, even though I don't really know exactly what individuation is supposed to be. |
| 18616 | If a term doesn't pick out a kind, keeping it may block improvements in classification [Machery] |
| Full Idea: If a hypothesised natural kind term fails to pick out a natural kind, keeping this theoretical term is likely to prevent the development of a new classification system that would identify the relevant kinds. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8.2.3) | |
| A reaction: I'm persuaded. This is why metaphysicians should stop talking about 'properties'. |
| 18614 | Vertical arguments say eliminate a term if it picks out different natural kinds in different theories [Machery] |
| Full Idea: Vertical arguments for eliminativism of theoretical terms note that distinct types of generalisation do not line up with each other. ...It is argued that the theoretical term picks out more than one natural kind. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8.2.3) | |
| A reaction: He mentions 'depression', as behavioural and cognitive; the former includes apes, and the latter doesn't. It is a nice principle for tidying up theories. |
| 18615 | Horizontal arguments say eliminate a term if it fails to pick out a natural kind [Machery] |
| Full Idea: Horizontal arguments for eliminativism of theoretical terms say that some terms should be eliminated if they do not pick out a natural kind. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8.2.3) | |
| A reaction: This is the one Machery likes, but I would say that it is less obvious than the 'vertical' version, since picking out a natural kind may not be the only job of a theoretical term. (p.238: Machery agrees!) |
| 18609 | Psychologists use 'induction' as generalising a property from one category to another [Machery] |
| Full Idea: Typically, psychologists use 'induction' to refer to the capacity to generalise a property from a category (the source) to another category (the target). | |
| From: Edouard Machery (Doing Without Concepts [2009], 7.1.1) | |
| A reaction: This is because psychologists are interested in the ongoing activities of thought. Philosophers step back a bit, to ask how the whole thing could get started. Philosophical induction has to start with individuals and single observations. |
| 18610 | 'Ampliative' induction infers that all members of a category have a feature found in some of them [Machery] |
| Full Idea: Induction is 'ampliative' when it infers that all or most members of a category possess a property from the fact that some of its members have this property. | |
| From: Edouard Machery (Doing Without Concepts [2009], 7.1.1) | |
| A reaction: This sounds like a simple step in reasoning, but actually it is more like explanation, and will involve overall coherence and probability, rather than a direct conclusion. This invites sceptical questions. The last one observed may be the exception. |
| 18562 | Connectionists cannot distinguish concept-memories from their background, or the processes [Machery] |
| Full Idea: Connectionists typically do not distinguish between processes and memory stores, and, more importantly, it is unclear whether connectionists can draw a distinction between the knowledge stored in a concept and the background. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.1) | |
| A reaction: In other words connectionism fails to capture the structured nature of our thinking. There is an innate structure (which, say I, should mainly be seen as 'mental files'). |
| 18561 | We can identify a set of cognitive capacities which are 'higher order' [Machery] |
| Full Idea: Categorization, deduction, induction, analogy-making, linguistic understanding, and planning - all of these are higher cognitive capacities. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.1) | |
| A reaction: His 'lower' competences are perceptual and motor. I say the entry to the higher competences are abstraction, idealisation and generalisation. If you can't do these (chimpanzees!) you will not be admitted. |
| 18574 | Concepts for categorisation and for induction may be quite different [Machery] |
| Full Idea: In general, concepts that are used when we categorise and concepts that are used when we reason inductively could have little in common. | |
| From: Edouard Machery (Doing Without Concepts [2009], 3.2.1) | |
| A reaction: In the end he is going to reject concepts altogether, so he would say this. Friends of concepts would be very surprised if the mind were so uneconomical in its activities, given that induction seems to be up to its neck in categorisation. |
| 18588 | Concept theories aim at their knowledge, processes, format, acquisition, and location [Machery] |
| Full Idea: A theory of concepts should determine the knowledge stored in them, and the cognitive processes that use concepts. Ideally it should also characterise their format, their acquisition, and (increasingly) localise them in the brain. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4) | |
| A reaction: Machery reveals his dubious scientism in the requirement to localise them in the brain. That strikes me as entirely irrelevant to both philosophy and psychology. I want the format, acquisition and knowledge. |
| 18611 | We should abandon 'concept', and just use 'prototype', 'exemplar' and 'theory' [Machery] |
| Full Idea: The notion of 'concept' ought to be eliminated from the theoretical vocabulary of psychology, and replaced by the notions of prototype, exemplar, and theory. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8) | |
| A reaction: Machery's main thesis. I think similarly about 'property' in metaphysics. It embraces different ideas, and if we eliminated 'property' (and used predicate, class, fundamental power, complex power) we would do better. Psychologists have dropped 'memory'. |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| Full Idea: Cantor (in his exploration of infinities) pushed the bounds of conceivability further than anyone before him. To discover what is conceivable, we have to enquire into the concept. | |
| From: report of George Cantor (works [1880]) by Michèle Friend - Introducing the Philosophy of Mathematics 6.5 | |
| A reaction: This remark comes during a discussion of Husserl's phenomenology. Intuitionists challenge Cantor's claim, and restrict what is conceivable to what is provable. Does possibility depend on conceivability? |
| 18567 | In the philosophy of psychology, concepts are usually introduced as constituents of thoughts [Machery] |
| Full Idea: In the philosophy of psychology, concepts are usually introduced as constituents, components, or parts of thoughts. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.4.3) | |
| A reaction: My instincts are against this. I take the fundamentals of concepts to be mental responses to distinct individual items in the world. Thought builds up from that. He says psychologists themselves don't see it this way. Influence of Frege. |
| 18569 | In philosophy theories of concepts explain how our propositional attitudes have content [Machery] |
| Full Idea: A philosophical theory of concepts is a semantic theory for our propositional attitudes: it explains how our thoughts can have the content they have. | |
| From: Edouard Machery (Doing Without Concepts [2009], 2.1.2) | |
| A reaction: I suppose this is what I am interested in. I want to know in what way concepts form a bridge between content and world. I am more interested in the propositions, and less interested in our attitudes towards them. |
| 18563 | By 'concept' psychologists mean various sorts of representation or structure [Machery] |
| Full Idea: Psychologists use 'concept' interchangeably with 'mental representation', 'category representation', 'knowledge representation', 'knowledge structure', 'semantic representation', and 'conceptual structures'. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.1) | |
| A reaction: [Machery gives references for each of these] Machery is moving in to attack these, but we look to psychologists to give some sort of account of what a concept might consist of, such that it could be implemented by neurons. |
| 18558 | Concept theorists examine their knowledge, format, processes, acquisition and location [Machery] |
| Full Idea: Psychological theories of concepts try to describe the knowledge stored in concepts, the format of concepts, the cognitive processes that use the concepts, the acquisition of concepts, and the localization of concepts in the brain. | |
| From: Edouard Machery (Doing Without Concepts [2009], Intro) | |
| A reaction: I suppose it would the first two that are of central interest. What individuates a concept (its 'format') and what are the contents of a concept. The word 'stored' seems to imply a mental files view. |
| 18557 | Psychologists treat concepts as long-term knowledge bodies which lead to judgements [Machery] |
| Full Idea: In psychology, concepts are characterized as those bodies of knowledge that are stored in long-term memory and used most higher cognitive competences when these processes result in judgements. | |
| From: Edouard Machery (Doing Without Concepts [2009], Intro) | |
| A reaction: Machery mounts an attack on this idea. I like the 'mental files' idea, where a concept starts as a label, and then acquires core knowledge, and then further information. The 'concept' is probably no more than a label, and minimal starter information. |
| 18560 | Psychologist treat concepts as categories [Machery] |
| Full Idea: Psychologists often use 'concept' and 'category' interchangeably. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.1) | |
| A reaction: Well they shouldn't. Some concepts are no more than words, and don't categorise anything. Some things may be categorised by a complex set of concepts. |
| 18592 | The concepts OBJECT or AGENT may be innate [Machery] |
| Full Idea: Several concepts, such as OBJECT or AGENT, may be innate. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.1.4) | |
| A reaction: It is one thing to say that we respond to objects and agents, and another to say that we have those 'concepts'. Presumably birds, and even bees, have to relate to similar features. Add PROCESS? |
| 18566 | Concepts should contain working memory, not long-term, because they control behaviour [Machery] |
| Full Idea: We ought to reserve the term 'concept' for the bodies of knowledge in working memory, and not for our knowledge of long-term memory, because the former, and not the latter, 'control behaviour'. | |
| From: Edouard Machery (Doing Without Concepts [2009], 1.4.1) | |
| A reaction: [He cites the psychologist Barsalou 1993] Some more theoretical concepts can only be recalled with difficulty, and control our theorising rather than our behaviour. But we act on some theories, so there is no clear borderline. |
| 18584 | One hybrid theory combines a core definition with a prototype for identification [Machery] |
| Full Idea: One hybrid theory of concepts says they have both a core and an identification procedure. The core is a definition (necessary and sufficient conditions), while the identification procedure consists of a prototype (the properties typical of a category). | |
| From: Edouard Machery (Doing Without Concepts [2009], 3.3.1) | |
| A reaction: This combines the classical and prototype theories of concepts. I like it because it fits the idea of 'mental files' nicely (see Recanati). If concepts are files (as in a database) they will have aspects like labels, basic info, and further details. |
| 18585 | Heterogeneous concepts might have conflicting judgements, where hybrid theories will not [Machery] |
| Full Idea: The Heterogeneity Hypothesis, but not the hybrid theory of concepts, predicts that the coreferential bodies of knowledge it posits will occasionally lead to conflicting outcomes, such as inconsistent judgements. | |
| From: Edouard Machery (Doing Without Concepts [2009], 3.3.2) | |
| A reaction: Machery's book champions the Heterogeneous Hypothesis. Hybrid views say the aspects of a concept are integrated, but Heterogeneity says there are separate processes. My preferred 'file' approach would favour integration. |
| 18578 | Concepts as definitions was rejected, and concepts as prototypes, exemplars or theories proposed [Machery] |
| Full Idea: Since the rejection of the classical theory of concepts (that they are definitions), three paradigms have successively emerged in the psychology of concepts: the prototype paradigm, the exemplar paradigm, and the theory paradigm. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4) | |
| A reaction: I am becoming a fan of the 'theory theory' proposal, because the concepts centre around what explains the phenomenon, which fits my explanatory account of essentialism. Not that it's right because it agrees with me, of course..... |
| 18575 | The concepts for a class typically include prototypes, and exemplars, and theories [Machery] |
| Full Idea: Across domains (such as biology and psychology) classes of physical objects, substances and events are typically represented by a prototype, by a set of exemplars, and by a theory. | |
| From: Edouard Machery (Doing Without Concepts [2009], 3.2.3) | |
| A reaction: In other words he thinks that all of the major psychological theories of concepts are partially correct, and he argues for extensive pluralism in the true picture. Bad news for neat philosophy, but real life is a right old mess. |
| 18591 | Classical theory can't explain facts like typical examples being categorised quicker [Machery] |
| Full Idea: The nail in the coffin of the classical theory is its lack of explanatory power. For example it doesn't explain the fact that typical x's are categorised more quickly and more reliably than atypical x's. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.1.3) | |
| A reaction: [He cites Rosch and Mervis: 1975:ch 5] This research launched the 'prototype' theory, which has since been challenged by the 'exemplar' and 'theory theory' rivals (and neo-empiricism, and idealisation). |
| 18583 | Many categories don't seem to have a definition [Machery] |
| Full Idea: For many categories there is simply no definition to learn (such as Wittgenstein's example of a 'game'). | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.1.4) |
| 18590 | Classical theory implies variety in processing times, but this does not generally occur [Machery] |
| Full Idea: If a concept is defined by means of another, such as MURDER by means of KILL, then processing the former concept should take longer in the classical theory, but several experiments show that this is not the case. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.1.3) | |
| A reaction: For the philosopher there is no escaping the findings of neuroscience when it comes to the study of concepts. This invites the question of the role, if any, of philosophy. I take philosophy to concern the big picture, or it is nothing. |
| 18594 | Knowing typical properties of things is especially useful in induction [Machery] |
| Full Idea: Knowing which properties are typical of a class is particularly useful when you have to draw inductions about the members of a class. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.2.1) |
| 18593 | The term 'prototype' is used for both typical category members, and the representation [Machery] |
| Full Idea: The term 'prototype' is used ambiguously to designate the most typical members of a category, and the representation of a category. (I use the term in the second sense). | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.2.1 n25) |
| 18595 | Prototype theories are based on computation of similarities with the prototype [Machery] |
| Full Idea: The most important property of prototype theories is that cognitive processes are assumed to involve the computation of the similarity between prototypes and other representations. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.2.3) | |
| A reaction: [He cites J.A.Hampton 1998, 2006] This presumably suits theories of the mind as largely computational (e.g. Fodor's account, based on the Turing machine). |
| 18596 | Prototype theorists don't tell us how we select the appropriate prototype [Machery] |
| Full Idea: We are typically not told how prototypes are selected, that is, what determines whether a specific prototype is retrieved from memory in order to be involved in the categorisation process. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.2.4) | |
| A reaction: One of the aims of this database is to make people aware of ideas that people have already thought of. This one was spotted 2,400 years ago. It's the Third Man problem. How do you even start to think about a particular thing? |
| 18603 | Maybe concepts are not the typical properties, but the ideal properties [Machery] |
| Full Idea: Barsalou (1983,1985) introduced the idea of ideals instead of prototypes. An ideal is a body of knowledge about the properties a thing should possess (rather than its typical actual properties). ... A 'bully' might be perfect, rather than typical. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.5.3) | |
| A reaction: [compressed] Machery offers this as an interesting minor variant, with little experimental support. I take idealisation to be one of the three key mental operations that enable us to think about the world (along with abstraction and generalisation). |
| 18605 | It is more efficient to remember the prototype, than repeatedly create it from exemplars [Machery] |
| Full Idea: Instead of regularly producing a prototype out of the exemplars stored in long-term memory, it seems more efficient to extract a prototype from category members during concept learning and to use this prototype when needed. | |
| From: Edouard Machery (Doing Without Concepts [2009], 6.3.2) | |
| A reaction: [This is a critique of Barsalou's on-the-fly proposal for prototypes] If the exemplar theory is right, then some sort of summary must occur when faced with a new instance. So this thought favours prototypes against exemplars. |
| 18606 | The prototype view predicts that typical members are easier to categorise [Machery] |
| Full Idea: The prototype paradigm of concepts makes the strong prediction that typical members should be easier to categorise than atypical members. | |
| From: Edouard Machery (Doing Without Concepts [2009], 6.4.1) | |
| A reaction: This is why philosophers should approach the topic of concepts with caution. Clearly empirical testing is going to settle this matter, not abstract theorising. |
| 18597 | Concepts as exemplars are based on the knowledge of properties of each particular [Machery] |
| Full Idea: The exemplar paradigm of concepts is built around the idea that concepts are sets of exemplars. In turn, an exemplar is a body of knowledge about the properties believed to be possessed by a particular member of a class. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.3.1) | |
| A reaction: I like the fact that this theory is rooted in particulars, where the prototype theory doesn't seem to say much about how prototypes are derived. But you have to do more than just contemplate a bunch of exemplars. |
| 18598 | Exemplar theories need to explain how the relevant properties are selected from a multitude of them [Machery] |
| Full Idea: Exemplar theories have a selection problem. Given that individuals have an infinite number of properties, they need to explain why exemplars represent such and such properties, instead of others. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.3.1) | |
| A reaction: I have the impression that this idea rests on the 'abundant' view of properties - that every true predicate embodies a property. A sparse view of properties might give a particular quite a restricted set of properties. |
| 18599 | In practice, known examples take priority over the rest of the set of exemplars [Machery] |
| Full Idea: An object that is extremely similar to a specific known category member, but only moderately similar to others, is more likely to be categorised as a category member than an object that is moderately similar to most known category members. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.3.3) | |
| A reaction: This research finding is a problem for the Exemplar Theory, in which all the exemplars have equal status. It is even a problem for the Prototype Theory, since the known member may not be like the prototype. |
| 18600 | Theory Theory says category concepts are knowledge stores explaining membership [Machery] |
| Full Idea: According to theory theorists, a concept of a category stores some knowledge that can explain the properties of the category members. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.4.1) | |
| A reaction: This is the account of essentialism which I defended in my PhD thesis. So naturally I embrace a theory of the nature of concepts which precisely dovetails with my view. I take explanation to be the central concept in metaphysics. |
| 18601 | Theory Theory says concepts are explanatory knowledge, and concepts form domains [Machery] |
| Full Idea: The two core ideas of the Theory Theory are that concepts are bodies of knowledge that underlie explanation, where explanation rests on folk examples, and concepts are organised in domains which use similar knowledge. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.4.1) | |
| A reaction: Folk explanation is opposed to scientific explanation, as expounded by Hempel etc. This sounds better and better, since the domains reflect the structure of reality. Machery defends Theory Theory as part of the right answer, but it's my favourite bit. |
| 18607 | Theory theorists rely on best explanation, rather than on similarities [Machery] |
| Full Idea: Theory theorists deny that categorisation depends on similarity; they often propose that categorisation involves some kind of inference to the best explanation. | |
| From: Edouard Machery (Doing Without Concepts [2009], 6.5.1) | |
| A reaction: Love it. Any theory of concepts should, in my view, be continuous with a plausible account of animal minds, and best explanations are not their strong suit. Maybe its explanations for slow categorising, and something else when it's quick. |
| 18608 | If categorisation is not by similarity, it seems to rely on what properties things might have [Machery] |
| Full Idea: It seems that when subjects are not categorising by similarity, they are relying on what properties objects can and cannot have - that is, on some modal knowledge. | |
| From: Edouard Machery (Doing Without Concepts [2009], 6.5.1) | |
| A reaction: I would call this essentialist categorisation, based on the inner causal powers which generate the modal profile of the thing. We categorise bullets and nails very differently, because of their modal profiles. |
| 18587 | The theory account is sometimes labelled as 'knowledge' or 'explanation' in approach [Machery] |
| Full Idea: The theory paradigm is sometimes called 'the knowledge approach' (Murphy 2002) or 'explanation-based views' (Komatsu 1992). | |
| From: Edouard Machery (Doing Without Concepts [2009], 4) | |
| A reaction: The word 'explanation' is music to my ears, so I am immediately sympathetic to the theory theory of concepts, even if it falls at the final hurdle. |
| 18577 | The word 'grandmother' may be two concepts, with a prototype and a definition [Machery] |
| Full Idea: If a prototype of grandmothers represents them as grey-haired old women, and a definition of grandmothers represents them as being necessarily the mother of a parent ....we may fail to recognise that 'grandmother' represents two distinct concepts. | |
| From: Edouard Machery (Doing Without Concepts [2009], 3.3.4) | |
| A reaction: He is referring to two distinct theories about what a concept is. He argues that both theories apply, so words do indeed represent several different concepts. Nice example. |
| 18589 | For behaviourists concepts are dispositions to link category members to names [Machery] |
| Full Idea: Behaviourists identified concepts with a mere disposition to associate category members with a given name. | |
| From: Edouard Machery (Doing Without Concepts [2009], 4.1.1) | |
| A reaction: This is one reason why the word 'disposition' triggers alarm bells in the immediately post-behaviourist generation of philosophers. The proposal is far too linguistic in character. |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| Full Idea: Cantor thought that we abstract a number as something common to all and only those sets any one of which has as many members as any other. ...However one wants to see the logic of the inference. The irony is that set theory lays out this logic. | |
| From: comment on George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: The logic Hart has in mind is the notion of an equivalence relation between sets. This idea sums up the older and more modern concepts of abstraction, the first as psychological, the second as logical (or trying very hard to be!). Cf Idea 9145. |
| 18612 | Americans are more inclined to refer causally than the Chinese are [Machery] |
| Full Idea: Tests suggest that American subjects were significantly more likely than Chinese subjects to have intuitions in line with causal-historical theories of reference. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8.1.3) | |
| A reaction: This is an example of 'experimental philosophy' in action (of which Machery is a champion). The underlying idea is that Americans are generally more disposed to think causally than the Chinese are. So more scientific? What do the Hopi do? |
| 18613 | Artifacts can be natural kinds, when they are the object of historical enquiry [Machery] |
| Full Idea: Some artifacts are the objects of inquiry in the social sciences ...such as prehistoric tools ...and hence, artifacts are bona fide natural kinds. | |
| From: Edouard Machery (Doing Without Concepts [2009], 8.2.1) | |
| A reaction: Presumably if a bird's nest can be a natural kind, then so can a flint axe, but then so can a mobile phone, for an urban anthropologist. 'Natural' is, to put it mildly, a tricky word. |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| Full Idea: Cantor proved that one-dimensional space has exactly the same number of points as does two dimensions, or our familiar three-dimensional space. | |
| From: report of George Cantor (works [1880]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.14 |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| Full Idea: Cantor said that only God is absolutely infinite. | |
| From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 1 | |
| A reaction: We are used to the austere 'God of the philosophers', but this gives us an even more austere 'God of the mathematicians'. |