80 ideas
| 13860 | We can only learn from philosophers of the past if we accept the risk of major misrepresentation [Wright,C] |
| Full Idea: We can learn from the work of philosophers of other periods only if we are prepared to run the risk of radical and almost inevitable misrepresentation of his thought. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Pref) | |
| A reaction: This sounds about right, and a motto for my own approach to Aristotle and Leibniz, but I see the effort as more collaborative than this suggests. Professional specialists in older philosophers are a vital part of the team. Read them! |
| 13883 | The best way to understand a philosophical idea is to defend it [Wright,C] |
| Full Idea: The most productive way in which to attempt an understanding of any philosophical idea is to work on its defence. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.vii) | |
| A reaction: Very nice. The key point is that this brings greater understanding than working on attacking an idea, which presumably has the dangers of caricature, straw men etc. It is the Socratic insight that dialectic is the route to wisdom. |
| 10142 | The attempt to define numbers by contextual definition has been revived [Wright,C, by Fine,K] |
| Full Idea: Frege gave up on the attempt to introduce natural numbers by contextual definition, but the project has been revived by neo-logicists. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction II |
| 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. |
| 9868 | An expression refers if it is a singular term in some true sentences [Wright,C, by Dummett] |
| Full Idea: For Wright, an expression refers to an object if it fulfils the 'syntactic role' of a singular term, and if we have fixed the truth-conditions of sentences containing it in such a way that some of them come out true. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michael Dummett - Frege philosophy of mathematics Ch.15 | |
| A reaction: Much waffle is written about reference, and it is nice to hear of someone actually trying to state the necessary and sufficient conditions for reference to be successful. So is it possible for 'the round square' to ever refer? '...is impossible to draw' |
| 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 Michčle 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 |
| 13861 | Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C] |
| Full Idea: In the Fregean view number theory is a science, aimed at those truths furnished by the essential properties of zero and its successors. The two broad question are then the nature of the objects, and the epistemology of those facts. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |
| A reaction: [compressed] I pounce on the word 'essence' here (my thing). My first question is about the extent to which the natural numbers all have one generic essence, and the extent to which they are individuals (bless their little cotton socks). |
| 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? |
| 13892 | One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C] |
| Full Idea: Someone could be clear about number identities, and distinguish numbers from other things, without conceiving them as ordered in a progression at all. The point of them would be to make comparisons between sizes of groups. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |
| A reaction: Hm. Could you grasp size if you couldn't grasp which of two groups was the bigger? What's the point of noting that I have ten pounds and you only have five, if you don't realise that I have more than you? You could have called them Caesar and Brutus. |
| 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. |
| 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. |
| 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). |
| 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 |
| 13867 | Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C] |
| Full Idea: The invitation to number the instances of some non-sortal concept is intelligible only if it is relativised to a sortal. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: I take this to be an essentially Fregean idea, as when we count the boots when we have decided whether they fall under the concept 'boot' or the concept 'pair'. I also take this to be the traditional question 'what units are you using'? |
| 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. |
| 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? |
| 17441 | Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck] |
| Full Idea: Wright is claiming that HP is a special sort of truth in some way: it is supposed to be the fundamental truth about cardinality; ...in particular, HP is supposed to be more fundamental, in some sense than the Dedekind-Peano axioms. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |
| A reaction: Heck notes that although PA can be proved from HP, HP can be proven from PA plus definitions, so direction of proof won't show fundamentality. He adds that Wright thinks HP is 'more illuminating'. |
| 13862 | There are five Peano axioms, which can be expressed informally [Wright,C] |
| Full Idea: Informally, Peano's axioms are: 0 is a number, numbers have a successor, different numbers have different successors, 0 isn't a successor, properties of 0 which carry over to successors are properties of all numbers. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |
| A reaction: Each statement of the famous axioms is slightly different from the others, and I have reworded Wright to fit him in. Since the last one (the 'induction axiom') is about properties, it invites formalization in second-order logic. |
| 17853 | Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C] |
| Full Idea: The intuitive proposal is the essential number theoretic truths are precisely the logical consequences of the Peano axioms, ...but the notion of consequence is a semantic one...and it is not obvious that we possess a semantic notion of the requisite kind. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |
| A reaction: (Not sure I understand this, but it is his starting point for rejecting PA as the essence of arithmetic). |
| 17854 | What facts underpin the truths of the Peano axioms? [Wright,C] |
| Full Idea: We incline to think of the Peano axioms as truths of some sort; so there has to be a philosophical question how we ought to conceive of the nature of the facts which make those statements true. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) | |
| A reaction: [He also asks about how we know the truths] |
| 13894 | Sameness of number is fundamental, not counting, despite children learning that first [Wright,C] |
| Full Idea: We teach our children to count, sometimes with no attempt to explain what the sounds mean. Doubtless it is this habit which makes it so natural to think of the number series as fundamental. Frege's insight is that sameness of number is fundamental. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |
| A reaction: 'When do children understand number?' rather than when they can recite numerals. I can't make sense of someone being supposed to understand number without a grasp of which numbers are bigger or smaller. To make 13='15' do I add or subtract? |
| 10140 | We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K] |
| Full Idea: Wright says the Fregean arithmetic can be broken down into two steps: first, Hume's Law may be derived from Law V; and then, arithmetic may be derived from Hume's Law without any help from Law V. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction I.4 | |
| A reaction: This sounds odd if Law V is false, but presumably Hume's Law ends up as free-standing. It seems doubtful whether the resulting theory would count as logic. |
| 8692 | Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend] |
| Full Idea: Wright proposed removing Frege's basic law V (which led to paradox), replacing it with Frege's 'number principle' (identity of numbers is one-to-one correspondence). The new system is formally consistent, and the Peano axioms can be derived from it. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michčle Friend - Introducing the Philosophy of Mathematics 3.7 | |
| A reaction: The 'number principle' is also called 'Hume's principle'. This idea of Wright's resurrected the project of logicism. The jury is ought again... Frege himself questioned whether the number principle was a part of logic, which would be bad for 'logicism'. |
| 17440 | Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck] |
| Full Idea: Wright intends the claim that Hume's Principle (HP) embodies an explanation of the concept of number to imply that it is analytic of the concept of cardinal number - so it is an analytic or conceptual truth, much as a definition would be. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 1 | |
| A reaction: Boolos is quoted as disagreeing. Wright is claiming a fundamental truth. Boolos says something can fix the character of something (as yellow fixes bananas), but that doesn't make it 'fundamental'. I want to defend 'fundamental'. |
| 13893 | It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C] |
| Full Idea: What is fundamental to possession of any notion of natural number at all is not the knowledge that the numbers may be arrayed in a progression but the knowledge that they are identified and distinguished by reference to 1-1 correlation among concepts. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xv) | |
| A reaction: My question is 'what is the essence of number?', and my inclination to disagree with Wright on this point suggests that the essence of number is indeed caught in the Dedekind-Peano axioms. But what of infinite numbers? |
| 13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C] |
| Full Idea: Identifying numbers with extensions will not solve the Caesar problem for numbers unless we have already solved the Caesar problem for extensions. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xiv) |
| 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). |
| 13869 | Number platonism says that natural number is a sortal concept [Wright,C] |
| Full Idea: Number-theoretic platonism is just the thesis that natural number is a sortal concept. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: See Crispin Wright on sortals to expound this. An odd way to express platonism, but he is presenting the Fregean version of it. |
| 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'. |
| 13870 | We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C] |
| Full Idea: We may not be able to settle whether some general form of empiricism is correct independently of natural numbers. It might be precisely our grasp of the abstract sortal, natural number, which shows the hypothesis of empiricism to be wrong. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: A nice turning of the tables. In the end only coherence decides these things. You may accept numbers and reject empiricism, and then find you have opened the floodgates for abstracta. Excessive floodgates, or blockages of healthy streams? |
| 13873 | Treating numbers adjectivally is treating them as quantifiers [Wright,C] |
| Full Idea: Treating numbers adjectivally is, in effect, treating the numbers as quantifiers. Frege observes that we can always parse out any apparently adjectival use of a number word in terms of substantival use. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iii) | |
| A reaction: The immediate response to this is that any substantival use can equally be expressed adjectivally. If you say 'the number of moons of Jupiter is four', I can reply 'oh, you mean Jupiter has four moons'. |
| 13899 | The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C] |
| Full Idea: The Peano Axioms are logical consequences of a statement constituting the core of an explanation of the notion of cardinal number. The infinity of cardinal numbers emerges as a consequence of the way cardinal number is explained. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xix) | |
| A reaction: This, along with Idea 13896, nicely summarises the neo-logicist project. I tend to favour a strategy which starts from ordering, rather than identities (1-1), but an attraction is that this approach is closer to counting objects in its basics. |
| 13896 | The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C] |
| Full Idea: We shall endeavour to see whether it is possible to follow through the strategy adumbrated in 'Grundlagen' for establishing the Peano Axioms without at any stage invoking classes. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi) | |
| A reaction: The key idea of neo-logicism. If you can avoid classes entirely, then set theory paradoxes become irrelevant, and classes aren't logic. Philosophers now try to derive the Peano Axioms from all sorts of things. Wright admits infinity is a problem. |
| 7804 | Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA] |
| Full Idea: Crispin Wright has reactivated Frege's logistic program, which for decades just about everybody assumed was a lost cause. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by José A. Benardete - Logic and Ontology 3 | |
| A reaction: [This opens Bernadete's section called "Back to Strong Logicism?"] |
| 13863 | Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C] |
| Full Idea: Most would cite Russell's paradox, the non-logical character of the axioms which Russell and Whitehead's reconstruction of Frege's enterprise was constrained to employ, and the incompleteness theorems of Gödel, as decisive for logicism's failure. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], Intro) |
| 13895 | The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C] |
| Full Idea: The general view is that Russell's Paradox put paid to Frege's logicist attempt, and Russell's own attempt is vitiated by the non-logical character of his axioms (esp. Infinity), and by the incompleteness theorems of Gödel. But these are bad reasons. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xvi) | |
| A reaction: Wright's work is the famous modern attempt to reestablish logicism, in the face of these objections. |
| 13884 | The idea that 'exist' has multiple senses is not coherent [Wright,C] |
| Full Idea: I have the gravest doubts whether any coherent account could be given of any multiplicity of senses of 'exist'. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 2.x) | |
| A reaction: I thoroughly agree with this thought. Do water and wind exist in different senses of 'exist'? |
| 13877 | Singular terms in true sentences must refer to objects; there is no further question about their existence [Wright,C] |
| Full Idea: When a class of terms functions as singular terms, and the sentences are true, then those terms genuinely refer. Being singular terms, their reference is to objects. There is no further question whether they really refer, and there are such objects. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iii) | |
| A reaction: This seems to be a key sentence, because this whole view is standardly called 'platonic', but it certainly isn't platonism as we know it, Jim. Ontology has become an entirely linguistic matter, but do we then have 'sakes' and 'whereaboutses'? |
| 9878 | Contextually defined abstract terms genuinely refer to objects [Wright,C, by Dummett] |
| Full Idea: Wright says we should accord to contextually defined abstract terms a genuine full-blown reference to objects. | |
| From: report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
| A reaction: This is the punch line of Wright's neo-logicist programme. See Idea 9868 for his view of reference. Dummett regards this strong view of contextual definition as 'exorbitant'. Wright's view strikes me as blatantly false. |
| 13868 | Sortal concepts cannot require that things don't survive their loss, because of phase sortals [Wright,C] |
| Full Idea: The claim that no concept counts as sortal if an instance of it can survive its loss, runs foul of so-called phase sortals like 'embryo' and 'chrysalis'. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: The point being that those items only fall under that sortal for one phase of their career, and of their identity. I've always thought such claims absurd, and this gives a good reason for my view. |
| 19508 | Contextualism needs a semantics for knowledge sentences that are partly indexical [Schiffer,S] |
| Full Idea: Contextualist semantics must capture the 'indexical' nature of knowledge claims, the fact that different utterances of a knowledge sentence with no apparent indexical terms can express different propositions. | |
| From: Stephen Schiffer (Contextualist Solutions to Scepticism [1996], p.325), quoted by Keith DeRose - The Case for Contextualism 1.5 | |
| A reaction: Schiffer tries to show that this is too difficult, and DeRose defends contextualism against the charge. |
| 19509 | The indexical aspect of contextual knowledge might be hidden, or it might be in what 'know' means [Schiffer,S] |
| Full Idea: One might have a 'hidden-indexical' theory of knowledge sentences: they contain constituents that are not the semantic values of any terms; ...or 'to know' itself might be indexical, as in 'I know[easy] I have hands' or 'I know[tough] I have hands'. | |
| From: Stephen Schiffer (Contextualist Solutions to Scepticism [1996], p.326-7), quoted by Keith DeRose - The Case for Contextualism 1.5 | |
| A reaction: [very compressed] Given the choice, I would have thought it was in 'know', since to say 'either you know p or you don't' sounds silly to me. |
| 13866 | A concept is only a sortal if it gives genuine identity [Wright,C] |
| Full Idea: Before we can conclude that φ expresses a sortal concept, we need to ensure that 'is the same φ as' generates statements of genuine identity rather than of some other equivalence relation. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) |
| 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 Michčle 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? |
| 13865 | 'Sortal' concepts show kinds, use indefinite articles, and require grasping identities [Wright,C] |
| Full Idea: A concept is 'sortal' if it exemplifies a kind of object. ..In English predication of a sortal concept needs an indefinite article ('an' elm). ..What really constitutes the distinction is that it involves grasping identity for things which fall under it. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.i) | |
| A reaction: This is a key notion, which underlies the claims of 'sortal essentialism' (see David Wiggins). |
| 13890 | Entities fall under a sortal concept if they can be used to explain identity statements concerning them [Wright,C] |
| Full Idea: 'Tree' is not a sortal concept under which directions fall since we cannot adequately explain the truth-conditions of any identity statement involving a pair of tree-denoting singular terms by appealing to facts to do with parallelism between lines. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 3.xiv) | |
| A reaction: The idea seems to be that these two fall under 'hedgehog', because that is a respect in which they are identical. I like to notion of explanation as a part of this. |
| 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. |
| 13898 | If we can establish directions from lines and parallelism, we were already committed to directions [Wright,C] |
| Full Idea: The fact that it seems possible to establish a sortal notion of direction by reference to lines and parallelism, discloses tacit commitments to directions in statements about parallelism...There is incoherence in the idea that a line might lack direction. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 4.xviii) | |
| A reaction: This seems like a slippery slope into a very extravagant platonism about concepts. Are concepts like direction as much a part of the natural world as rivers are? What other undiscovered concepts await us? |
| 13882 | A milder claim is that understanding requires some evidence of that understanding [Wright,C] |
| Full Idea: A mild version of the verification principle would say that it makes sense to think of someone as understanding an expression only if he is able, by his use of the expression, to give the best possible evidence that he understands it. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.vii) | |
| A reaction: That doesn't seem to tell us what understanding actually consists of, and may just be the truism that to demonstrate anything whatsoever will necessarily involve some evidence. |
| 13885 | If apparent reference can mislead, then so can apparent lack of reference [Wright,C] |
| Full Idea: If the appearance of reference can be misleading, why cannot an apparent lack of reference be misleading? | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 2.xi) | |
| A reaction: A nice simple thought. Analytic philosophy has concerned itself a lot with sentences that seem to refer, but the reference can be analysed away. For me, this takes the question of reference out of the linguistic sphere, which wasn't Wright's plan. |
| 17857 | We can accept Frege's idea of object without assuming that predicates have a reference [Wright,C] |
| Full Idea: The heart of the problem is Frege's assumption that predicates have Bedeutungen at all; and no reason is at present evident why someone who espouses Frege's notion of object is contrained to make that assumption. | |
| From: Crispin Wright (Frege's Concept of Numbers as Objects [1983], 1.iv) | |
| A reaction: This seems like a penetrating objection to Frege's view of reference, and presumably supports the Kripke approach. |
| 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'. |