67 ideas
6947 | Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce] |
Full Idea: Metaphysical systems have not usually rested upon any observed facts, or not in any great degree. They are chiefly adopted because their fundamental propositions seem 'agreeable to reason', which means that which we find ourselves inclined to believe. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.15) | |
A reaction: This leads to Peirce's key claim - that we should allow our beliefs to be formed by something outside of ourselves. I don't share Peirce's contempt for metaphysics, which I take to be about the most abstract presuppositions of our ordinary beliefs. |
6937 | Reason aims to discover the unknown by thinking about the known [Peirce] |
Full Idea: The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 7) | |
A reaction: I defy anyone to come up with a better definition of reasoning than that. The emphasis is on knowledge rather than truth, which you would expect from a pragmatist. …Actually the definition doesn't cover conditional reasoning terribly well. |
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. |
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 |
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. |
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. |
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? |
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'. |
21492 | Realism is basic to the scientific method [Peirce] |
Full Idea: The fundamental hypothesis of the method of science is this: There are real things, whose characters are entirely independent of our opinion of them. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Albert Atkin - Peirce 3 'method' | |
A reaction: He admits later that this is only a commitment and not a fact. It seems to me that when you combine this idea with the huge success of science, the denial of realism is crazy. Philosophy has a lot to answer for. |
6949 | If someone doubted reality, they would not actually feel dissatisfaction [Peirce] |
Full Idea: Nobody can really doubt that there are Reals, for, if he did, doubt would not be a source of dissatisfaction. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.19) | |
A reaction: This rests on Peirce's view that all that really matters is a sense of genuine dissatisfaction, rather than a theoretical idea. So even at the end of Meditation One, Descartes isn't actually worried about whether his furniture exists. |
15083 | The fundamental case of logical necessity is the valid conclusion of an inference [McFetridge, by Hale] |
Full Idea: McFetridge's conception of logical necessity is one which sees the concept as receiving its fundamental exemplification in the connection between the premiss and conclusion of a deductively valid inference. | |
From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
A reaction: This would mean that p could be logically necessary but false (if it was a valid argument from false premisses). What if it was a valid inference in a dodgy logical system (including 'tonk', for example)? |
15084 | In the McFetridge view, logical necessity means a consequent must be true if the antecedent is [McFetridge, by Hale] |
Full Idea: McFetridge's view proves that if the conditional corresponding to a valid inference is logically necessary, then there is no sense in which it is possible that its antecedent be true but its consequent false. ..This result generalises to any statement. | |
From: report of Ian McFetridge (Logical Necessity: Some Issues [1986]) by Bob Hale - Absolute Necessities 2 | |
A reaction: I am becoming puzzled by Hale's assertion that logical necessity is 'absolute', while resting his case on a conditional. Are we interested in the necessity of the inference, or the necessity of the consequent? |
12181 | Traditionally, logical necessity is the strongest, and entails any other necessities [McFetridge] |
Full Idea: The traditional crucial assumption is that logical necessity is the strongest notion of necessity. If it is logically necessary that p, then it is necessary that p in any other use of the notion of necessity there may be (physically, practically etc.). | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: Sounds right. We might say it is physically necessary simply because it is logically necessary, and even that it is metaphysically necessary because it is logically necessary (required by logic). Logical possibility is hence the weakest kind? |
12184 | Logical necessity overrules all other necessities [McFetridge] |
Full Idea: If it is logically necessary that if p then q, then there is no other sense of 'necessary' in which it is not necessary that if p then q. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: The thesis which McFetridge proposes to defend. The obvious rival would be metaphysical necessity, and the rival claim would presumably be that things are only logically necessary if that is entailed by a metaphysical necessity. Metaphysics drives logic. |
12180 | Logical necessity requires that a valid argument be necessary [McFetridge] |
Full Idea: There will be a legitimate notion of 'logical' necessity only if there is a notion of necessity which attaches to the claim, concerning a deductively valid argument, that if the premisses are true then so is the conclusion. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: He quotes Aristotle's Idea 11148 in support. Is this resting a stronger idea on a weaker one? Or is it the wrong way round? We endorse validity because we see the necessity; we don't endorse necessity because we see 'validity'. |
12183 | It is only logical necessity if there is absolutely no sense in which it could be false [McFetridge] |
Full Idea: Is there any sense in which, despite an ascription of necessity to p, it is held that not-p is possible? If there is, then the original claim then it was necessary is not a claim of 'logical' necessity (which is the strongest necessity). | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: See Idea 12181, which leads up to this proposed "test" for logical necessity. McFetridge has already put epistemic ('for all I know') possibility to one side. □p→¬◊¬p is the standard reading of necessity. His word 'sense' bears the burden. |
12192 | The mark of logical necessity is deduction from any suppositions whatever [McFetridge] |
Full Idea: The manifestation of the belief that a mode of inference is logically necessarily truth-preserving is the preparedness to employ that mode of inference in reasoning from any set of suppositions whatsoever. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §4) | |
A reaction: He rests this on the idea of 'cotenability' of the two sides of a counterfactual (in Mill, Goodman and Lewis). There seems, at first blush, to be a problem of the relevance of the presuppositions. |
12182 | We assert epistemic possibility without commitment to logical possibility [McFetridge] |
Full Idea: Time- and person-relative epistemic possibility can be asserted even when logical possibility cannot, such as undecided mathematical propositions. 'It may be that p' just comes to 'For all I know, not-p'. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §1) | |
A reaction: If it is possible 'for all I know', then it could be actual for all I know, and if we accept that it might be actual, we could hardly deny that it is logically possible. Logical and epistemic possibilities of mathematical p stand or fall together. |
12187 | Objectual modal realists believe in possible worlds; non-objectual ones rest it on the actual world [McFetridge] |
Full Idea: The 'objectual modal realist' holds that what makes modal beliefs true are certain modal objects, typically 'possible worlds'. ..The 'non-objectual modal realist' says modal judgements are made true by how things stand with respect to this world. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
A reaction: I am an enthusiastic 'non-objectual modal realist'. I accept the argument that real possible worlds have no relevance to the actual world, and explain nothing (see Jubien). The possibilities reside in the 'powers' of this world. See Molnar on powers. |
12186 | Modal realists hold that necessities and possibilities are part of the totality of facts [McFetridge] |
Full Idea: The 'modal realist' holds that part of the totality of what is the case, the totality of facts, are such things as that certain events could have happened, certain propositions are necessarily true, if this happened then that would have been the case. | |
From: Ian McFetridge (Logical Necessity: Some Issues [1986], §2) | |
A reaction: I am an enthusiastic modal realist. If the aim of philosophy is 'to understand' (and I take that to be the master idea of the subject) then no understanding is possible which excludes the possibilities and necessities in things. |
6941 | We are entirely satisfied with a firm belief, even if it is false [Peirce] |
Full Idea: As soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
A reaction: This does not deny that the truth or falsehood of a belief is independent of whether we are satisfied with it. It is making a fair point, though, about why we believe things, and it can't be because of truth, because we don't know how to ensure that. |
6940 | The feeling of belief shows a habit which will determine our actions [Peirce] |
Full Idea: The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions. Doubt never has such an effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.10) | |
A reaction: It is one thing to assert this fairly accurate observation, and another to assert that this is the essence or definition of a belief. Perhaps it is the purpose of belief, without being the phenomenological essence of it. We act in states of uncertainty. |
6942 | We want true beliefs, but obviously we think our beliefs are true [Peirce] |
Full Idea: We seek for a belief that we shall think to be true; but we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: If, as I do, you like to define belief as 'commitment to truth', Peirce makes a rather startling observation. You are rendered unable to ask whether your beliefs are true, because you have defined them as true. Nice point… |
6943 | A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce] |
Full Idea: The mere putting of a proposition into the interrogative form does not stimulate the mind to any struggle after belief; there must be a real and living doubt. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: This the attractive aspect of Peirce's pragmatism, that he is always focusing on real life rather than abstract theory or pure logic. |
6598 | We need our beliefs to be determined by some external inhuman permanency [Peirce] |
Full Idea: It is necessary that a method should be found by which our beliefs be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877]), quoted by Robert Fogelin - Walking the Tightrope of Reason Ch.5 | |
A reaction: This very sensible and interesting remark hovers somewhere between empiricism and pragmatism. Fogelin very persuasively builds his account of knowledge on it. The key point is that we hardly ever choose what to believe. See Idea 2454. |
6944 | Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce] |
Full Idea: It is a common idea that demonstration must rest on indubitable propositions, either first principles of a general nature, or first sensations; but actual demonstration is completely satisfactory if it starts from propositions free from all actual doubt. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: Another nice example of Peirce focusing on the practical business of thinking, rather than abstract theory. I agree with this approach, that explanation and proof do not aim at perfection and indubitability, but at what satisfies a critical mind. |
6948 | Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce] |
Full Idea: To satisfy our doubts it is necessary that a method should be found by which our beliefs may be determined by nothing human, but by some external permanency - by something upon which our thinking has no effect. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.18) | |
A reaction: This may be the single most important idea in pragmatism and in the philosophy of science. See Fodor on experiments (Idea 2455). Put the question to nature. The essential aim is to be passive in our beliefs - just let reality form them. |
6945 | Once doubt ceases, there is no point in continuing to argue [Peirce] |
Full Idea: Some people seem to love to argue a point after all the world is fully convinced of it. But no further advance can be made. When doubt ceases, mental action on the subject comes to an end; and, if it did go on, it would be without purpose. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.11) | |
A reaction: This is the way Peirce's pragmatism, which deals with how real thinking actually works (rather than abstract logic), deals with scepticism. However, there is a borderline where almost everyone is satisfied, but the very wise person remains sceptical. |
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. |
6939 | What is true of one piece of copper is true of another (unlike brass) [Peirce] |
Full Idea: The guiding principle is that what is true of one piece of copper is true of another; such a guiding principle with regard to copper would be much safer than with regard to many other substances - brass, for example. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
A reaction: Peirce is so beautifully simple and sensible. This gives the essential notion of a natural kind, and is a key notion in our whole understanding of physical reality. |
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 |
6938 | Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce] |
Full Idea: It is probably of more advantage to an animal to have his mind filled with pleasing and encouraging visions, independently of their truth; and thus, upon unpractical subjects, natural selection might occasion a fallacious tendency of thought. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p. 8) | |
A reaction: Note that this is a pragmatist saying that a set of beliefs might work fine but be untrue. So Peirce does not have the highly relativistic notion of truth of some later pragmatists. Good for him. Note the early date to be thinking about Darwin. |
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'. |
6946 | If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce] |
Full Idea: If death is annihilation, then the man who believes that he will certainly go straight to heaven when he dies, provided he have fulfilled certain simple observances in this life, has a cheap pleasure which will not be followed by the least disappointment. | |
From: Charles Sanders Peirce (The Fixation of Belief [1877], p.12) | |
A reaction: This is a nicely wicked summary of one side of Pascal's options. All the problems of the argument are built into Peirce's word "cheap". Peirce goes on to talk about ostriches burying their heads. |