89 ideas
19635 | Hegel produced modern optimism; he failed to grasp that consciousness never progresses [Hegel, by Cioran] |
Full Idea: Hegel is chiefly responsible for modern optimism. How could he have failed to see that consciousness changes only its forms and modalities, but never progresses. | |
From: report of Georg W.F.Hegel (works [1812]) by E.M. Cioran - A Short History of Decay 5 |
8215 | Hegel was the last philosopher of the Book [Hegel, by Derrida] |
Full Idea: Hegel was the last philosopher of the Book. | |
From: report of Georg W.F.Hegel (works [1812]) by Jacques Derrida - Positions p.64 | |
A reaction: Reference to 'the Book' connects this to the great religions which rely on one holy text. The implication is that Hegel was proposing one big solution to all problems. It is doubtful if many philosophers before Hegel dreamt of that either. |
16011 | Hegel doesn't storm the heavens like the giants, but works his way up by syllogisms [Kierkegaard on Hegel] |
Full Idea: Hegel is a Johannes Climacus who does not storm the heavens, like the giants, by putting mountain upon mountain, but climbs aboard them by way of his syllogisms. | |
From: comment on Georg W.F.Hegel (works [1812]) by Søren Kierkegaard - The Journals of Kierkegaard 2A | |
A reaction: [Idea from SY] This appears to be an attempt at insulting Hegel for his timidity, but it seems to be describing the cautious approach which most modern philosophers take to be correct. [PG] |
5433 | For Hegel, things are incomplete, and contain external references in their own nature [Hegel, by Russell] |
Full Idea: The basis of Hegel's system is that what is incomplete must not be self-subsistent, and needs the support of other things; whatever has relations to things outside itself must contain some reference to those outside things in its own nature. | |
From: report of Georg W.F.Hegel (works [1812]) by Bertrand Russell - Problems of Philosophy Ch.14 | |
A reaction: This leads to the idealist doctrine of 'internal relations'. It has some plausibility if you think about the physicist's definition of mass, which has to refer to forces etc. Presumably there is one essence for all of reality, instead of separate ones. |
3301 | On the continent it is generally believed that metaphysics died with Hegel [Benardete,JA on Hegel] |
Full Idea: In continental Europe it is widely believed that the metaphysical game was played out in Hegel. | |
From: comment on Georg W.F.Hegel (works [1812]) by José A. Benardete - Metaphysics: the logical approach Intro |
14456 | 'Socrates is human' expresses predication, and 'Socrates is a man' expresses identity [Russell] |
Full Idea: The is of 'Socrates is human' expresses the relation of subject and predicate; the is of 'Socrates is a man' expresses identity. It is a disgrace to the human race that it employs the same word 'is' for these entirely different ideas. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
A reaction: Does the second one express identity? It sounds more like membership to me. 'Socrates is the guy with the hemlock' is more like identity. |
19661 | Making sufficient reason an absolute devalues the principle of non-contradiction [Hegel, by Meillassoux] |
Full Idea: Hegel saw that the absolutization of the principle of sufficient reason (which marked the culmination of the belief in the necessity of what is) required the devaluation of the principle of non-contradiction. | |
From: report of Georg W.F.Hegel (works [1812], 3) by Quentin Meillassoux - After Finitude; the necessity of contingency 3 | |
A reaction: I pass this on without understanding it, though a joint study of my collection of ideas on sufficient reason and non-contradiction might make it clear. [Let me know if you can explain it!] |
20952 | Rather than in three stages, Hegel presented his dialectic as 'negation of the negation' [Hegel, by Bowie] |
Full Idea: Hegel's 'dialectic' is often characterised in terms of the triad of thesis, antithesis and synthesis. This is, however, not the way he presents it. The core of the dialectic is rather what Hegel terms the 'negation of the negation'. | |
From: report of Georg W.F.Hegel (works [1812]) by Andrew Bowie - Introduction to German Philosophy | |
A reaction: Interestingly, this connects it to debates about intuitionist logic, which denies that double-negation necessarily makes a positive. Presumably Marx emphasised the first reading. |
14426 | A definition by 'extension' enumerates items, and one by 'intension' gives a defining property [Russell] |
Full Idea: The definition of a class or collection which enumerates is called a definition by 'extension', and one which mentions a defining property is called a definition by 'intension'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: In ordinary usage we take intensional definitions for granted, so it is interesting to realise that you might define 'tiger' by just enumerating all the tigers. But all past tigers? All future tigers? All possible tigers which never exist? |
8468 | The sentence 'procrastination drinks quadruplicity' is meaningless, rather than false [Russell, by Orenstein] |
Full Idea: Russell proposed (in his theory of types) that sentences like 'The number two is fond of cream cheese' or 'Procrastination drinks quadruplicity' should be regarded as not false but meaningless. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
A reaction: This seems to be the origin of the notion of a 'category mistake', which Ryle made famous. The problem is always poetry, where abstractions can be reified, or personified, and meaning can be squeezed out of almost anything. |
14454 | An argument 'satisfies' a function φx if φa is true [Russell] |
Full Idea: We say that an argument a 'satisfies' a function φx if φa is true. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
A reaction: We end up with Tarski defining truth in terms of satisfaction, so we shouldn't get too excited about what he achieved (any more than he got excited). |
14453 | The Darapti syllogism is fallacious: All M is S, all M is P, so some S is P' - but if there is no M? [Russell] |
Full Idea: Some moods of the syllogism are fallacious, e.g. 'Darapti': 'All M is S, all M is P, therefore some S is P', which fails if there is no M. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
A reaction: This critique rests on the fact that the existential quantifier entails some existence, but the universal quantifier does not. |
14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
Full Idea: We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here. |
14428 | Members define a unique class, whereas defining characteristics are numerous [Russell] |
Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) |
14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
Full Idea: There is no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VIII) |
14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
Full Idea: The Axiom of Infinity may be enunciated as 'If n be any inductive cardinal number, there is at least one class of individuals having n terms'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIII) | |
A reaction: So for every possible there exists a set of terms for it. Notice that they are 'terms', not 'objects'. We must decide whether we are allowed terms which don't refer to real objects. |
14443 | The British parliament has one representative selected from each constituency [Russell] |
Full Idea: We have a class of representatives, who make up our Parliament, one being selected out of each constituency. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: You can rely on Russell for the clearest illustrations of these abstract ideas. He calls the Axiom of Choice the 'Multiplicative' Axiom. |
14444 | Choice is equivalent to the proposition that every class is well-ordered [Russell] |
Full Idea: Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class). | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: Russell calls Choice the 'Multiplicative' Axiom. |
14445 | Choice shows that if any two cardinals are not equal, one must be the greater [Russell] |
Full Idea: The [Axiom of Choice] is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: It is illuminating for the uninitiated to learn that this result can't be taken for granted (with infinite cardinals). |
14446 | We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell] |
Full Idea: Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: A deservedly famous illustration of a rather tricky part of set theory. |
14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
Full Idea: The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true? | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: I take this to say that in the theory of types, it is possible to reduce each level of type down to one type. |
14461 | Propositions about classes can be reduced to propositions about their defining functions [Russell] |
Full Idea: It is right (in its main lines) to say that there is a reduction of propositions nominally about classes to propositions about their defining functions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: The defining functions will involve the theory of types, in order to avoid the paradoxes of naïve set theory. This is Russell's strategy for rejecting the existence of sets. |
8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
Full Idea: Russell's solution (in the theory of types) consists of restricting the principle that every predicate has a set as its extension so that only meaningful predicates have sets as their extensions. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
A reaction: There might be a chicken-and-egg problem here. How do you decide the members of a set (apart from ostensively) without deciding the predicate(s) that combine them? |
8745 | Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell] |
Full Idea: The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2 | |
A reaction: I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way. |
14452 | All the propositions of logic are completely general [Russell] |
Full Idea: It is part of the definition of logic that all its propositions are completely general. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) |
14462 | In modern times, logic has become mathematical, and mathematics has become logical [Russell] |
Full Idea: Logic has become more mathematical, and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
A reaction: This appears to be true even if you reject logicism about mathematics. Logicism is sometimes rejected because it always ends up with a sneaky ontological commitment, but maybe mathematics shares exactly the same commitment. |
12444 | Logic is concerned with the real world just as truly as zoology [Russell] |
Full Idea: Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
A reaction: I love this idea and am very sympathetic to it. The rival view seems to be that logic is purely conventional, perhaps defined by truth tables etc. It is hard to see how a connective like 'tonk' could be self-evidently silly if it wasn't 'unnatural'. |
10057 | Logic can only assert hypothetical existence [Russell] |
Full Idea: No proposition of logic can assert 'existence' except under a hypothesis. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
A reaction: I am prepared to accept this view fairly dogmatically, though Musgrave shows some of the difficulties of the if-thenist view (depending on which 'order' of logic is being used). |
14464 | Logic can be known a priori, without study of the actual world [Russell] |
Full Idea: Logical propositions are such as can be known a priori, without study of the actual world. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
A reaction: This remark constrasts strikingly with Idea 12444, which connects logic to the actual world. Is it therefore a priori synthetic? |
8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
21777 | Negation of negation doubles back into a self-relationship [Hegel, by Houlgate] |
Full Idea: For Hegel, the 'negation of negation' is negation that, as it were, doubles back on itself and 'relates itself to itself'. | |
From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 6 'Space' | |
A reaction: [ref VNP 1823 p.108] Glad we've cleared that one up. |
10450 | Russell admitted that even names could also be used as descriptions [Russell, by Bach] |
Full Idea: Russell clearly anticipated Donnellan when he said proper names can also be used as descriptions, adding that 'there is nothing in the phraseology to show whether they are being used in this way or as names'. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.175) by Kent Bach - What Does It Take to Refer? 22.2 L1 | |
A reaction: This seems also to anticipate Strawson's flexible and pragmatic approach to these things, which I am beginning to think is correct. |
14457 | Names are really descriptions, except for a few words like 'this' and 'that' [Russell] |
Full Idea: We can even say that, in all such knowledge as can be expressed in words, with the exception of 'this' and 'that' and a few other words of which the meaning varies on different occasions - no names occur, but what seem like names are really descriptions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
A reaction: I like the caveat about what is expressed in words. Russell is very good at keeping non-verbal thought in the picture. This is his famous final reduction of names to simple demonstratives. |
14458 | Asking 'Did Homer exist?' is employing an abbreviated description [Russell] |
Full Idea: When we ask whether Homer existed, we are using the word 'Homer' as an abbreviated description. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
A reaction: It is hard to disagree with Russell over this rather unusual example. It doesn't seem so plausible when Ottiline refers to 'Bertie'. |
7311 | The only genuine proper names are 'this' and 'that' [Russell] |
Full Idea: In all knowledge that can be expressed in words - with the exception of "this" and "that", and a few other such words - no genuine proper names occur, but what seem like genuine proper names are really descriptions | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) | |
A reaction: This is the terminus of Russell's train of thought about descriptions. Suppose you point to something non-existent, like a ghost in a misty churchyard? You'd be back to the original problem of naming a non-existent! |
14455 | 'I met a unicorn' is meaningful, and so is 'unicorn', but 'a unicorn' is not [Russell] |
Full Idea: In 'I met a unicorn' the four words together make a significant proposition, and the word 'unicorn' is significant, …but the two words 'a unicorn' do not form a group having a meaning of its own. It is an indefinite description describing nothing. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVI) |
14442 | If straight lines were like ratios they might intersect at a 'gap', and have no point in common [Russell] |
Full Idea: We wish to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a 'gap' and have no point in common. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], X) | |
A reaction: You can make a Dedekind Cut in the line of ratios (the rationals), so there must be gaps. I love this idea. We take for granted intersection at a point, but physical lines may not coincide. That abstract lines might fail also is lovely! |
14438 | New numbers solve problems: negatives for subtraction, fractions for division, complex for equations [Russell] |
Full Idea: Every generalisation of number has presented itself as needed for some simple problem. Negative numbers are needed to make subtraction always possible; fractions to make division always possible; complex numbers to make solutions of equations possible. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) | |
A reaction: Doesn't this rather suggest that we made them up? If new problems turn up, we'll invent another lot. We already have added 'surreal' numbers. |
8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
13510 | Could a number just be something which occurs in a progression? [Russell, by Hart,WD] |
Full Idea: Russell toyed with the idea that there is nothing to being a natural number beyond occurring in a progression | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.8) by William D. Hart - The Evolution of Logic 5 | |
A reaction: How could you define a progression, without a prior access to numbers? - Arrange all the objects in the universe in ascending order of mass. Use scales to make the selection. Hence a finite progression, with no numbers! |
18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
A reaction: The sequence is 'Cauchy' if N exists. |
14436 | A series can be 'Cut' in two, where the lower class has no maximum, the upper no minimum [Russell] |
Full Idea: There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. This division of a series into two classes is called a 'Dedekind Cut'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
14439 | A complex number is simply an ordered couple of real numbers [Russell] |
Full Idea: A complex number may be regarded and defined as simply an ordered couple of real numbers | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VII) |
14421 | Discovering that 1 is a number was difficult [Russell] |
Full Idea: The discovery that 1 is a number must have been difficult. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
A reaction: Interesting that he calls it a 'discovery'. I am tempted to call it a 'decision'. |
14424 | Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell] |
Full Idea: We want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) | |
A reaction: Why would just having certain formal properties be insufficient for counting? You just need an ordered series of unique items. It isn't just that we 'want' this. If you define something that we can't count with, you haven't defined numbers. |
14441 | The formal laws of arithmetic are the Commutative, the Associative and the Distributive [Russell] |
Full Idea: The usual formal laws of arithmetic are the Commutative Law [a+b=b+a and axb=bxa], the Associative Law [(a+b)+c=a+(b+c) and (axb)xc=ax(bxc)], and the Distributive Law [a(b+c)=ab+ac)]. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IX) |
14420 | Infinity and continuity used to be philosophy, but are now mathematics [Russell] |
Full Idea: The nature of infinity and continuity belonged in former days to philosophy, but belongs now to mathematics. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Pref) | |
A reaction: It is hard to disagree, since mathematicians since Cantor have revealed so much about infinite numbers (through set theory), but I think it remains an open question whether philosophers have anything distinctive to contribute. |
8764 | Categories are the best foundation for mathematics [Shapiro] |
Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
14431 | The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell] |
Full Idea: Order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. ...Without it we would not be able to define the order of magnitude among fractions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], IV) |
14423 | '0', 'number' and 'successor' cannot be defined by Peano's axioms [Russell] |
Full Idea: That '0', 'number' and 'successor' cannot be defined by means of Peano's five axioms, but must be independently understood. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) |
14422 | Any founded, non-repeating series all reachable in steps will satisfy Peano's axioms [Russell] |
Full Idea: Given any series which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], I) |
14425 | A number is something which characterises collections of the same size [Russell] |
Full Idea: The number 3 is something which all trios have in common, and which distinguishes them from other collections. A number is something that characterises certain collections, namely, those that have that number. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: This is a verbal summary of the Fregean view of numbers, which marks the arrival of set theory as the way arithmetic will in future be characterised. The question is whether set theory captures all aspects of numbers. Does it give a tool for counting? |
8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
14434 | What matters is the logical interrelation of mathematical terms, not their intrinsic nature [Russell] |
Full Idea: What matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
A reaction: If they have an instrinsic nature, that would matter far more, because that would dictate the interrelations. Structuralism seems to require that they don't actually have any intrinsic nature. |
8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
14465 | Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell] |
Full Idea: 'Ten men' is grammatically the same form as 'white men', so that 10 might be thought to be an adjective qualifying 'men'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
A reaction: The immediate problem, as Frege spotted, is that such expressions can be rephrased to remove the adjective (by saying 'the number of men is ten'). |
13414 | For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf] |
Full Idea: Russell's own stand was that numbers are really only sets of equivalent sets. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Paul Benacerraf - Logicism, Some Considerations (PhD) p.168 | |
A reaction: Benacerraf is launching a nice attack on this view, based on our inability to grasp huge numbers on this basis, or to see their natural order. |
8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
14449 | There is always something psychological about inference [Russell] |
Full Idea: There is always unavoidably something psychological about inference. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
A reaction: Glad to find Russell saying that. Only pure Fregeans dream of a logic that rises totally above the minds that think it. See Robert Hanna on the subject. |
14463 | Existence can only be asserted of something described, not of something named [Russell] |
Full Idea: Existence can only be asserted of something described, not of something named. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVIII) | |
A reaction: This is the motivation behind Russell's theory of definite descriptions, and epitomises the approach to ontology through language. Sounds wrong to me! |
5645 | The dialectical opposition of being and nothing is resolved in passing to the concept of becoming [Hegel, by Scruton] |
Full Idea: The concept of being contains within itself it own negation - nothing - and the dialectical opposition between these two concepts is resolved only in the passage to a new concept, becoming, which contains the truth of the passage from nothing to being. | |
From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Short History of Modern Philosophy Ch.12 | |
A reaction: The idea that one concept 'contains' another, or that an opposition could be 'resolved' by a new concept, sounds doubtful to me. For most analytical philosophers, and for Aristotle, oppositions are contradictions, and cannot and should not be 'resolved'. |
5646 | Hegel gives an ontological proof of the existence of everything [Hegel, by Scruton] |
Full Idea: It would not be unfair to say that Hegel's metaphysics consists of an ontological proof of the existence of everything. | |
From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Short History of Modern Philosophy Ch.12 | |
A reaction: This is so gloriously far from David Hume that we must all find some appeal in it. The next question would be whether necessary existence has been proved. If so, given death, decay and entropy, what is it that has to exist? 2nd Law of Thermodynamics? |
14429 | Classes are logical fictions, made from defining characteristics [Russell] |
Full Idea: Classes may be regarded as logical fictions, manufactured out of defining characteristics. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II n1) | |
A reaction: I agree with this. The idea that in addition to the members there is a further object, the set containing them, is absurd. Sets are a tool for thinking about the world. |
21755 | For Hegel, categories shift their form in the course of history [Hegel, by Houlgate] |
Full Idea: For Hegel, the categories of thought are not fixed, eternal forms that remain unchanged throughout history, but are concepts that alter their meaning in history. | |
From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
A reaction: This results from a critique of Kant's rather rigid view of categories. This idea is very influential, and certainly counts among Hegel's better ideas. |
21754 | Our concepts and categories disclose the world, because we are part of the world [Hegel, by Houlgate] |
Full Idea: For Hegel, the structure of our concepts and categories is identical with, and thus discloses, the structure of the world itself, because we ourselves are born into and so share the character of the world we encounter. | |
From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
A reaction: This is a reasonable speculation, but it makes more sense in the context of natural selection, and an empiricist theory of concepts. |
22079 | Hegel said Kant's fixed categories actually vary with culture and era [Hegel, by Houlgate] |
Full Idea: Hegel's disagreement with Kant is that categories are not unambiguously universal forms of human understanding, but are conceived in subtly different ways in different cultures and in different historical epochs. | |
From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - Hegel p.95 | |
A reaction: This may be Hegel's most influential idea. Though he hoped that categories would contain truth, by arising untrammelled from reason, and thereby matching reality. His successors seem to have given up on that hope, and settled for relativism. |
14430 | If a relation is symmetrical and transitive, it has to be reflexive [Russell] |
Full Idea: It is obvious that a relation which is symmetrical and transitive must be reflexive throughout its domain. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: Compare Idea 13543! The relation will return to its originator via its neighbours, rather than being directly reflexive? |
14432 | 'Asymmetry' is incompatible with its converse; a is husband of b, so b can't be husband of a [Russell] |
Full Idea: The relation of 'asymmetry' is incompatible with the converse. …The relation 'husband' is asymmetrical, so that if a is the husband of b, b cannot be the husband of a. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], V) | |
A reaction: This is to be contrasted with 'non-symmetrical', where there just happens to be no symmetry. |
14435 | The essence of individuality is beyond description, and hence irrelevant to science [Russell] |
Full Idea: The essence of individuality always eludes words and baffles description, and is for that very reason irrelevant to science. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
A reaction: [context needed for a full grasp of this idea] Russell seems to refer to essence as much as to individuality. The modern essentialist view is that essences are not beyond description after all. Fundamental physics is clearer now than in 1919. |
12197 | Inferring q from p only needs p to be true, and 'not-p or q' to be true [Russell] |
Full Idea: In order that it be valid to infer q from p, it is only necessary that p should be true and that the proposition 'not-p or q' should be true. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
A reaction: Rumfitt points out that this approach to logical consequences is a denial of any modal aspect, such as 'logical necessity'. Russell observes that for a good inference you must know the disjunction as a whole. Could disjunction be modal?... |
14450 | All forms of implication are expressible as truth-functions [Russell] |
Full Idea: There is no need to admit as a fundamental notion any form of implication not expressible as a truth-function. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIV) | |
A reaction: Note that this is from a book about 'mathematical' philosophy. Nevertheless, it seems to have the form of a universal credo for Russell. He wasn't talking about conditionals here. Maybe conditionals are not implications (in isolation, that is). |
14460 | If something is true in all possible worlds then it is logically necessary [Russell] |
Full Idea: Saying that the axiom of reducibility is logically necessary is what would be meant by saying that it is true in all possible worlds. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: This striking remark is a nice bridge between Leibniz (about whom Russell wrote a book) and Kripke. |
9225 | Hegel reputedly claimed to know a priori that there are five planets [Hegel, by Field,H] |
Full Idea: Hegel is reputed to have claimed to have deduced on a priori grounds that the number of planets is exactly five. | |
From: report of Georg W.F.Hegel (works [1812]) by Hartry Field - Recent Debates on the A Priori 1 | |
A reaction: Even if this is a wicked travesty of Hegel, it will do nicely to represent the extremes of claims to a priori synthetic knowledge. Field doesn't offer any evidence. I would love it to be true. |
8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
14433 | Mathematically expressed propositions are true of the world, but how to interpret them? [Russell] |
Full Idea: We know that certain scientific propositions - often expressed in mathematical symbols - are more or less true of the world, but we are very much at sea as to the interpretation to be put upon the terms which occur in these propositions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VI) | |
A reaction: Enter essentialism, say I! Russell's remark is pretty understandable in 1919, but I don't think the situation has changed much. The problem of interpretation may be of more interest to philosophers than to physicists. |
14451 | Propositions are mainly verbal expressions of true or false, and perhaps also symbolic thoughts [Russell] |
Full Idea: We mean by 'proposition' primarily a form of words which expresses what is either true or false. I say 'primarily' because I do not wish to exclude other than verbal symbols, or even mere thoughts if they have a symbolic character. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XV) | |
A reaction: I like the last bit, as I think of propositions as pre-verbal thoughts, and I am sympathetic to Fodor's 'language of thought' thesis, that there is a system of representations within the brain. |
21758 | Humans have no fixed identity, but produce and reveal their shifting identity in history [Hegel, by Houlgate] |
Full Idea: For Hegel, the absolute truth of humanity is that human beings have no fixed, given identity, but rather determine and produce their own identity and their world in history, and that they gradually come to the recognition of this fact in history. | |
From: report of Georg W.F.Hegel (works [1812]) by Stephen Houlgate - An Introduction to Hegel 01 | |
A reaction: This quintessentially existentialist idea, most obvious in Sartre, seems to have originated with this view of Hegel's. |
20414 | Hegel's Absolute Spirit is the union of human rational activity at a moment, and whatever that sustains [Hegel, by Eldridge] |
Full Idea: We may take Hegel's Absolute Spirit to be the union of collective, human rational activity at a historical moment with its proper object, the forms of social and individual life that the rational activity is devoted to understanding and sustaining. | |
From: report of Georg W.F.Hegel (works [1812]) by Richard Eldridge - G.W.F. Hegel (aesthetics) 1 | |
A reaction: From this formulation it sounds as if the whole human race might have momentary union, but presumably it is more local 'peoples' that can exhibit this. |
3909 | Society isn’t founded on a contract, since contracts presuppose a society [Hegel, by Scruton] |
Full Idea: For Hegel, society cannot be founded on a contract, since contracts have no reality until society is in place. | |
From: report of Georg W.F.Hegel (works [1812]) by Roger Scruton - Modern Philosophy:introduction and survey 28.2 | |
A reaction: Interesting, and reminiscent of the private language argument, but contracts surely start as deals between individuals (on a desert island?). |
4347 | When man wills the natural, it is no longer natural [Hegel] |
Full Idea: When man wills the natural, it is no longer natural. | |
From: Georg W.F.Hegel (works [1812]), quoted by Rosalind Hursthouse - On Virtue Ethics Ch.4 | |
A reaction: Sounds good, though I'm not sure what it means. The application of the word 'natural' seems a bit arbitrary to me. No objective joint exists between the natural and unnatural. The default position has to be that everything is natural. |
4188 | Hegel's entire philosophy is nothing but a monstrous amplification of the ontological proof [Schopenhauer on Hegel] |
Full Idea: Hegel's entire philosophy is nothing but a monstrous amplification of the ontological proof. | |
From: comment on Georg W.F.Hegel (works [1812]) by Arthur Schopenhauer - Abstract of 'The Fourfold Root' Ch.II | |
A reaction: All massive a priori metaphysics is summed up in this argument, which is right at the core of philosophy. |
6686 | Hegel said he was offering an encyclopaedic rationalisation of Christianity [Hegel, by Graham] |
Full Idea: Hegel claimed that his philosophy was nothing less than an encyclopaedic rationalisation of the Christian religion. | |
From: report of Georg W.F.Hegel (works [1812]) by Gordon Graham - Eight Theories of Ethics Ch.5 | |
A reaction: Why did he pick Christianity to rationalise? How can you reason properly if you start with a dogma? |