75 ideas
| 4767 | Traditionally, rational beliefs are those which are justified by reasons [Psillos] |
| Full Idea: The traditional conception of Reason is that all beliefs should be justified (that is, backed up by reasons) in order to be rational. | |
| From: Stathis Psillos (Causation and Explanation [2002], §1.6) | |
| A reaction: I think it is the duty of all philosophers to either defend this traditional view, or quit philosophy for some other activity. Rorty suggests hermeneutics. In a democracy, rulers should be continually required to give reasons for their decisions. |
| 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 |
| 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). |
| 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 |
| 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. |
| 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] |
| 4810 | Valid deduction is monotonic - that is, it remains valid if further premises are added [Psillos] |
| Full Idea: Valid deductive arguments have the property of monotonicity; if the conclusion Q follows from the premises P, then it will also follow if further premises P* are added to P. | |
| From: Stathis Psillos (Causation and Explanation [2002], §9.2.1) | |
| A reaction: For perversity's sake we could add a new premise which contradicted one of the original ones ('Socrates is a god'). Or one premise could be 'I believe..', and the new one could show that the belief was false. Induction is non-monotonic. |
| 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. |
| 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 |
| 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 |
| 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 |
| 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. |
| 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 |
| 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. |
| 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. |
| 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. |
| 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? |
| 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 |
| 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'. |
| 4768 | The 'epistemic fallacy' is inferring what does exist from what can be known to exist [Psillos] |
| Full Idea: The move from what can (or cannot) be known to exist to what does (or does not) exist has been dubbed the 'epistemic fallacy'. | |
| From: Stathis Psillos (Causation and Explanation [2002], §1.6) | |
| A reaction: This should be a standard concept in all philosophical discussion. It is the commonest, simplest, and most profound blunder made by philosophers, and they do it all the time. |
| 4808 | If we say where Mars was two months ago, we offer an explanation without a prediction [Psillos] |
| Full Idea: There can be explanations without predictions, as when we explain a previous position of Mars from its present one, plus a law. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.9) | |
| A reaction: If we don't mind stretching the word, I think we can 'predict' the past, as where I predict the location of an Egyptian tomb from my study of papyruses. |
| 4807 | A good barometer will predict a storm, but not explain it [Psillos] |
| Full Idea: There can be predictions without explanations, as when a barometer successfully predicts storms, but on its own it does not explain them. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.8) | |
| A reaction: Actually, barometers contribute to explanations. A reasonable predictor might offer no explanation ('if he's out, she's probably out too'), but an infallible predictor is almost certain to involve causation, which helps a lot in explanation. |
| 4811 | Induction (unlike deduction) is non-monotonic - it can be invalidated by new premises [Psillos] |
| Full Idea: Unlike deductive arguments, induction is non-monotonic - that is, it can be invalidated by the addition of new premises. | |
| From: Stathis Psillos (Causation and Explanation [2002], §9.2.1) | |
| A reaction: This is a fancy way of stating the obvious, which is that induction is not a type of deduction. Hume is sometimes accused of this false assumption. Presumably induction is rational, even if it is not actually logical. |
| 4812 | Explanation is either showing predictability, or showing necessity, or showing causal relations [Psillos] |
| Full Idea: The three types of explanation are 'epistemic' (the event is expectable because of a law), or 'modal' (the event is necessary because of a law), or 'ontic' (it is shown how the event fits into the world's causal structure). | |
| From: Stathis Psillos (Causation and Explanation [2002], §11.1) | |
| A reaction: Prediction, necessity or causes. It is hard to think of any other way to explain something. Presumably you would exclude necessities if you didn't believe in them. Hume would go for prediction, on the basis of regularities. Personally, I want it all. |
| 4802 | Just citing a cause does not enable us to understand an event; we also need a relevant law [Psillos] |
| Full Idea: Explanation has to do with understanding; just citing a cause would not offer an adequate understanding, unless it was accompanied by the citation of a law that connects the two events. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.2) | |
| A reaction: It is surely undeniable that being told the cause but not the law will increase our understanding. Understanding and explanation come in degrees. Full understanding would require an explanation of the law, and beyond. Any relevant truth helps. |
| 4804 | The 'covering law model' says only laws can explain the occurrence of single events [Psillos] |
| Full Idea: The 'deductive-nomological' model became known as the 'covering law model': its main thesis is that laws and only laws adequately explain the occurrence of singular events. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.2) | |
| A reaction: But presumably you need other events to derive a law, so you could say that a singular event can only be explained if it isn't singular. A regularity pattern would offer a partial explanation, before any law had been derived. |
| 4805 | If laws explain the length of a flagpole's shadow, then the shadow also explains the length of the pole [Psillos] |
| Full Idea: If we use geometry and optics to explain the length of shadow cast by a flag-pole, this seems to be reversible, so that the shadow will explain the length of the pole. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.5) | |
| A reaction: A neat example which presumably implies that an explanation must involve temporal statements. The laws of physics are totally reversible in time, and so will not suffice to explain events on their own. Time's arrow becomes an axiom of explanation? |
| 4395 | There are non-causal explanations, most typically mathematical explanations [Psillos] |
| Full Idea: There are non-causal explanations, most typically mathematical explanations. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: A crucial basic point, which must be drummed into the minds of ruthless Quinean naturalists, who want to explain everything by quarks and electrons |
| 4806 | An explanation can just be a 'causal story', without laws, as when I knock over some ink [Psillos] |
| Full Idea: Knocking over an ink bottle explains the stain on the carpet, and it is not in doubt because you cannot quote the laws involved; a 'causal story' can give a complete explanation without a law. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.6) | |
| A reaction: But why is he so clumsy, and the bottle so unstable? Was it really (Freudian) an 'accident'? There is no end to complete explanation. But 'I was clumsy this once' and 'I am always clumsy' are equally good explanations. |
| 4404 | Maybe explanation is entirely relative to the interests and presuppositions of the questioner [Psillos] |
| Full Idea: Some philosophers focus on the so-called 'pragmatics of explanation' - that an explanation is an answer to a 'why' question, and the relevant answer will depend on the presuppositions or interests of the questioner. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: This seems to me right. Explanation is an entirely human business, not a feature of nature, and most explanations will track back to the big bang if you have the patience, but they always terminate because of pragmatic considerations. But fobbing off? |
| 4803 | An explanation is the removal of the surprise caused by the event [Psillos] |
| Full Idea: An explanation amounts to the removal of the initial surprise that accompanied the occurrence of the event. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.2) | |
| A reaction: This is a nice simple point. It makes explanation relative. God requires no explanations, small children require many. The implication is that explanations make events predictable, which means they must either offer inductive generalisations, or laws. |
| 4769 | It is hard to analyse causation, if it is presupposed in our theory of the functioning of the mind [Psillos] |
| Full Idea: There is a problem if causation is the object of our analysis, but is also presupposed (as an empirical principle of human psychology) for the functioning of the mind. | |
| From: Stathis Psillos (Causation and Explanation [2002], §1.7) | |
| A reaction: This doesn't sound like a major problem. If it is, it is presumably impossible to analyse the mind, because a mind is presupposed in the process of analysis. |
| 16002 | The self is a combination of pairs of attributes: freedom/necessity, infinite/finite, temporal/eternal [Kierkegaard] |
| Full Idea: A human being is essentially spirit, but what is spirit? Spirit is to be a self. But what is the Self? In short, it is a synthesis of the infinite and the finite, of the temporal and the eternal, of freedom and necessity. | |
| From: Søren Kierkegaard (Sickness unto Death [1849], p.59) | |
| A reaction: The dense language of his first paragraph was to poke fun at fashionable Hegelian writing. The book gets very lucid afterwards! [SY] |
| 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? |
| 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. |
| 4770 | Nothing is more usual than to apply to external bodies every internal sensation which they occasion [Psillos] |
| Full Idea: Nothing is more usual than to apply to external bodies every internal sensation which they occasion. | |
| From: Stathis Psillos (Causation and Explanation [2002], §1.8) | |
| A reaction: This is the core of Hume's is/ought claim - what he calls the mind 'spreading itself'. It is a powerful claim. Personally I think we have become TOO sceptical here, and have the delusion that crucial features of nature are created within our minds. |
| 4399 | Causes clearly make a difference, are recipes for events, explain effects, and are evidence [Psillos] |
| Full Idea: The platitudes of causation are that 1) causes make a difference (counterfactually or probabilistically), 2) causes are recipes for events, 3) causes explain their effects, and 4) causes are evidence for effects. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: A nice piece of analysis which offers some problems for anyone (like Russell) who wants to analyse causation completely out of our conceptual scheme. |
| 4400 | Theories of causation are based either on regularity, or on intrinsic relations of properties [Psillos] |
| Full Idea: While Humeans base their theories on the intuition of regularity, their opponents base theirs on the intuition that there is an intrinsic relation between the properties of two particular things involved (like a hammer and a vase). | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: I favour the intrinsic relation of properties view, but this leaves the question of whether we can explain a relation, apart from observing the regularities associated with the properties. |
| 4403 | We can't base our account of causation on explanation, because it is the wrong way round [Psillos] |
| Full Idea: We cannot distinguish between good and bad explanations of some phenomena, unless we first distinguish between causal and non-causal explanations. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: This seems right, but it pushes us towards the idea that causation is non-analysable, and must be taken as a metaphysically basic axiom. If naturalistic accounts fail, that may be only alternative. |
| 4789 | Three divisions of causal theories: generalist/singularist, intrinsic/extrinsic, reductive/non-reductive [Psillos] |
| Full Idea: The three ways to divide theories on causation are: between generalist and singularist, between intrinsic and extrinsic characterisations of the causal relationship, and between reductive and non-reductive approaches. | |
| From: Stathis Psillos (Causation and Explanation [2002], §4.5) | |
| A reaction: Okay. I vote for singularist, intrinsic and reductive. I'm guessing that that pushes me towards Salmon and Dowe's theory of the 'transfer of conserved quantities', which is certainly reductive, doesn't need regularities in the events, and seems intrinsic. |
| 4790 | If causation is 'intrinsic' it depends entirely on the properties and relations of the cause and effect [Psillos] |
| Full Idea: If causation is taken to be an 'intrinsic' relation, then that c causes e will have to depend entirely on the properties of c and e, and the relations between c and e. | |
| From: Stathis Psillos (Causation and Explanation [2002], §4.5.2) | |
| A reaction: This view would move us towards 'essentialism', that the essences of objects produce the events and the laws, rather than external imposed forces and laws. |
| 4402 | Empiricists tried to reduce causation to explanation, which they reduced to logic-plus-a-law [Psillos] |
| Full Idea: The logical empiricists (esp. Hempel) analysed the concept of causation in terms of causal explanation, and analysed the latter as a species of deductive argument, with one premises stating a universal law (the so-called Deductive-Nomological model). | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: This feels wrong, as deduction seems insufficiently naturalistic, and the assumption of a law as premise seems to beg heaps of questions. |
| 4774 | Counterfactual claims about causation imply that it is more than just regular succession [Psillos] |
| Full Idea: If counterfactual claims can be made about causation, this suggests that there is more to it than mere regular succession. | |
| From: Stathis Psillos (Causation and Explanation [2002], §2.2) | |
| A reaction: Interesting. Even Hume makes counterfactual claims in his first definition of cause, and all claims of causation seem to go beyond the immediate evidence. |
| 4793 | "All gold cubes are smaller than one cubic mile" is a true universal generalisation, but not a law [Psillos] |
| Full Idea: The statement "all gold cubes are smaller than one cubic mile" seems to have all the features demanded of a lawlike statement, yet it can hardly be said to express a law. It is a merely true universal generalisation. | |
| From: Stathis Psillos (Causation and Explanation [2002], §5.3) | |
| A reaction: Nice example. A trickier case is "all cubes of uranium are smaller than one cubic mile", which sounds like part of a law. It suggests a blurred borderline between the two. How much gold is there in the universe? Is that fact a natural necessity? |
| 4397 | Regularity doesn't seem sufficient for causation [Psillos] |
| Full Idea: A rather important objection to Humeanism has been that regularity is not sufficient for causation. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: Obviously a crucial problem, but the Humean view can defend itself by introducing other constant conjunctions. We don't observe events in isolation, but as part of a pattern of regularities. |
| 4401 | It is not a law of nature that all the coins in my pocket are euros, though it is a regularity [Psillos] |
| Full Idea: It is not a law of nature that all the coins in my pocket are euros, though it is a regularity. | |
| From: Stathis Psillos (Causation and Explanation [2002], Intro) | |
| A reaction: Good example, but it doesn't demolish the regularity view. We should come to conscious minds last. There aren't many other unfailing regularities that are not laws. |
| 4792 | A Humean view of causation says it is regularities, and causal facts supervene on non-causal facts [Psillos] |
| Full Idea: The Humean view depends on the conjunction of two general theses: first, causation is tied to regularity; secondly, causal facts supervene on non-causal facts. | |
| From: Stathis Psillos (Causation and Explanation [2002], §4.5.4) | |
| A reaction: If causation is just regularities, this means it is patterns observed by us, which means causation doesn't actually exist. So Hume is wrong. Singular causation is possible, and needs explanation. |
| 4801 | The regularity of a cock's crow is used to predict dawn, even though it doesn't cause it [Psillos] |
| Full Idea: A regularity can be used to predict a future event irrespective of whether it is deemed causal or not. A farmer can predict that dawn has broken on hearing the cock's crow. | |
| From: Stathis Psillos (Causation and Explanation [2002], §8.1) | |
| A reaction: This seems a highly significant criticism of any view that says regularity leads to causation, which is the basis of induction, which leads to counterfactual claims, and thus arrives a the laws of nature. |
| 4796 | Laws are sets of regularities within a simple and strong coherent system of wider regularities [Psillos] |
| Full Idea: In the 'web-of-laws' approach, laws are those regularities that are members of a coherent system of regularities, in particular, a system that can be represented as a deductive axiomatic system, striking a good balance between simplicity and strength. | |
| From: Stathis Psillos (Causation and Explanation [2002], §5.6) | |
| A reaction: Psillos attribute this view to Mill, Ramsey and Lewis. It is the obvious candidate for a fully developed Humean empiricist system, where regularities reinforce one another. I think laws are found in mechanisms, not in regularities, which are symptoms. |
| 4799 | Dispositional essentialism can't explain its key distinction between essential and non-essential properties [Psillos] |
| Full Idea: Many philosophers will find dispositional essentialism unappealing, not least because it seems to fail to explain how (and in virtue of what) there is this supposed fundamental distinction between essential and non-essential properties. | |
| From: Stathis Psillos (Causation and Explanation [2002]) | |
| A reaction: Maybe there is no precise definition, but any idiot can see that some properties of gold are essential (mass) and others non-essential (attractive to jackdaws). It's a fair question, but is this the strongest objection to essentialism? |
| 4780 | In some counterfactuals, the counterfactual event happens later than its consequent [Psillos] |
| Full Idea: In "had the acrobat jumped, there would have been a safety net" the antecedent of the counterfactual (the jumping) is temporally later than the consequent (the installation of the net). | |
| From: Stathis Psillos (Causation and Explanation [2002], §3.3) | |
| A reaction: This blocks anyone (e.g. David Lewis) who tries to define counterfactual claims entirely in terms of a condition followed by a consequence. Nice example. |
| 4791 | Counterfactual theories say causes make a difference - if c hadn't occurred, then e wouldn't occur [Psillos] |
| Full Idea: The counterfactual theory is a non-Humean relation between singular events; the thought is that causation makes a difference - to say that c causes e is to say that if c hadn't occurred, e wouldn't have occurred either. | |
| From: Stathis Psillos (Causation and Explanation [2002], §4.5.4) | |
| A reaction: Helpful. I'm beginning to think that this theory is wrong. It gives an account of how we see causation, and a test for it, but it says nothing about what causation actually is. |
| 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'. |